Is the function PARSENAME() the opposite of QUOTENAME()
Based on the comment by ypercubeᵀᴹ which reads:


  ...can you find a string x that parsename(quotename(x),1) will be different than x? If not, that falls into the definition of "inverse function" ;)  ...


... I created a simple statement to have a look at the different values returned by these functions. The code is as follows:

Code

DECLARE 
-- Text that I will be converting using QUOTENAME()
@Text2Quote_1 AS NVARCHAR(20),
@Text2Quote_2 AS NVARCHAR(20),
@Text2Quote_3 AS NVARCHAR(20),
@Text2Quote_4 AS NVARCHAR(20),
-- The characters used for "quotation"
@QuoteChar_1 AS NCHAR(1),
@QuoteChar_2 AS NCHAR(1),
@QuoteChar_3 AS NCHAR(1),
-- The Parsing option as defined in the original MS documnenation for PARSSENAME()
@ParseParam AS INT

SET @Text2Quote_1 = N'Test'
SET @Text2Quote_2 = N'[Test]'
SET @Text2Quote_3 = N'Test.dbo.test'
SET @Text2Quote_4 = N'[Test].[dbo].[test]'

SET @QuoteChar_1 = ''''
SET @QuoteChar_2 = '[' 
SET @QuoteChar_3 = '"' 

SET @ParseParam = 1 -- Parsing level : Object name
-- checking the results for ' single quotes
SELECT  @Text2Quote_1 AS OriginalText_1,
        @Text2Quote_2 AS OriginalText_2,
        @Text2Quote_3 AS OriginalText_3,
        @Text2Quote_4 AS OriginalText_4
SELECT  QUOTENAME(@Text2Quote_1,@QuoteChar_1) AS QuotedText_1, 
        QUOTENAME(@Text2Quote_2,@QuoteChar_1) AS QuotedText_2, 
        QUOTENAME(@Text2Quote_3,@QuoteChar_1) AS QuotedText_3, 
        QUOTENAME(@Text2Quote_4,@QuoteChar_1) AS QuotedText_4
select  PARSENAME(QUOTENAME(@Text2Quote_1,@QuoteChar_1),@ParseParam) AS Parsed_QuotedText_1, 
        PARSENAME(QUOTENAME(@Text2Quote_2,@QuoteChar_1),@ParseParam) AS Parsed_QuotedText_2,
        PARSENAME(QUOTENAME(@Text2Quote_3,@QuoteChar_1),@ParseParam) AS Parsed_QuotedText_3,
        PARSENAME(QUOTENAME(@Text2Quote_4,@QuoteChar_1),@ParseParam) AS Parsed_QuotedText_4


Basically, I set a string values, output that value, QUOTENAME() and output the value, then PARSENAME() the QUOTENAME()ed value and output it.

Results

The results were quite interesting:

OriginalText_1           OriginalText_2           OriginalText_3           OriginalText_4          
------------------------ ------------------------ ------------------------ ------------------------
Test                     [Test]                   Test.dbo.test            [Test].[dbo].[test]     

(1 row(s) affected)                                                                                

QuotedText_1             QuotedText_2             QuotedText_3             QuotedText_4            
------------------------ ------------------------ ------------------------ ------------------------
'Test'                   '[Test]'                 'Test.dbo.test'          '[Test].[dbo].[test]'   

(1 row(s) affected)                                                                                

Parsed_QuotedText_1      Parsed_QuotedText_2      Parsed_QuotedText_3      Parsed_QuotedText_4     
------------------------ ------------------------ ------------------------ ------------------------
'Test'                   NULL                     test'                    NULL                    

(1 row(s) affected)                                                                                


When the base text Test.dbo.test (OriginalText_3) is passed on to the QUOTENAME() function it gets converted to: 'Test.dbo.test' (QuotedText_3)

When the QuotedText_3 string 'Test.dbo.test' is passed on to the PARSENAME() function it is converted to: test' (Parsed_QuotedText_3)

Conclusion

Seeing as I have proven ypercubeᵀᴹ's thesis wrong, I think it is safe to state that the function PARSENAME() is not the inverse of QUOTENAME().



Based on Aaron Bertrand's feedback here the code for square brackets [ ] (add to original script)

Additional Code for square brackets [ ]

-- checking the results for [ brackets                                                                           
SELECT  @Text2Quote_1 AS OriginalText_1,                                                                         
        @Text2Quote_2 AS OriginalText_2,                                                                         
        @Text2Quote_3 AS OriginalText_3,                                                                         
        @Text2Quote_4 AS OriginalText_4                                                                          
SELECT  QUOTENAME(@Text2Quote_1,@QuoteChar_2) AS QuotedText_1,                             
        QUOTENAME(@Text2Quote_2,@QuoteChar_2) AS QuotedText_2,                             
        QUOTENAME(@Text2Quote_3,@QuoteChar_2) AS QuotedText_3,                             
        QUOTENAME(@Text2Quote_4,@QuoteChar_2) AS QuotedText_4                              
select  PARSENAME(QUOTENAME(@Text2Quote_1,@QuoteChar_2),@ParseParam) AS Parsed_QuotedText_1,
        PARSENAME(QUOTENAME(@Text2Quote_2,@QuoteChar_2),@ParseParam) AS Parsed_QuotedText_2,
        PARSENAME(QUOTENAME(@Text2Quote_3,@QuoteChar_2),@ParseParam) AS Parsed_QuotedText_3,
        PARSENAME(QUOTENAME(@Text2Quote_4,@QuoteChar_2),@ParseParam) AS Parsed_QuotedText_4


Output for square brackets [ ]

OriginalText_1           OriginalText_2           OriginalText_3           OriginalText_4
------------------------ ------------------------ ------------------------ ------------------------
Test                     [Test]                   Test.dbo.test            [Test].[dbo].[test]

(1 row(s) affected)

QuotedText_1             QuotedText_2             QuotedText_3             QuotedText_4
------------------------ ------------------------ ------------------------ ------------------------
[Test]                   [[Test]]]                [Test.dbo.test]          [[Test]].[dbo]].[test]]]

(1 row(s) affected)

Parsed_QuotedText_1      Parsed_QuotedText_2      Parsed_QuotedText_3      Parsed_QuotedText_4
------------------------ ------------------------ ------------------------ ------------------------
Test                     [Test]                   Test.dbo.test            [Test].[dbo].[test]

(1 row(s) affected)


and to complete things the code for the double quotes

Additional code for double quotes "

-- checking the results for [ brackets                                                                           
SELECT  @Text2Quote_1 AS OriginalText_1,                                                                         
        @Text2Quote_2 AS OriginalText_2,                                                                         
        @Text2Quote_3 AS OriginalText_3,                                                                         
        @Text2Quote_4 AS OriginalText_4                                                                          
SELECT  QUOTENAME(@Text2Quote_1,@QuoteChar_3) AS QuotedText_1,                             
        QUOTENAME(@Text2Quote_2,@QuoteChar_3) AS QuotedText_2,                             
        QUOTENAME(@Text2Quote_3,@QuoteChar_3) AS QuotedText_3,                             
        QUOTENAME(@Text2Quote_4,@QuoteChar_3) AS QuotedText_4                              
select  PARSENAME(QUOTENAME(@Text2Quote_1,@QuoteChar_3),@ParseParam) AS Parsed_QuotedText_1,
        PARSENAME(QUOTENAME(@Text2Quote_2,@QuoteChar_3),@ParseParam) AS Parsed_QuotedText_2,
        PARSENAME(QUOTENAME(@Text2Quote_3,@QuoteChar_3),@ParseParam) AS Parsed_QuotedText_3,
        PARSENAME(QUOTENAME(@Text2Quote_4,@QuoteChar_3),@ParseParam) AS Parsed_QuotedText_4


Output for double quotes "

OriginalText_1           OriginalText_2           OriginalText_3           OriginalText_4
------------------------ ------------------------ ------------------------ ------------------------
Test                     [Test]                   Test.dbo.test            [Test].[dbo].[test]

(1 row(s) affected)

QuotedText_1             QuotedText_2             QuotedText_3             QuotedText_4
------------------------ ------------------------ ------------------------ ------------------------
"Test"                   "[Test]"                 "Test.dbo.test"          "[Test].[dbo].[test]"

(1 row(s) affected)

Parsed_QuotedText_1      Parsed_QuotedText_2      Parsed_QuotedText_3      Parsed_QuotedText_4
------------------------ ------------------------ ------------------------ ------------------------
Test                     [Test]                   Test.dbo.test            [Test].[dbo].[test]

(1 row(s) affected)




Based on ypercube™'s feedback and Aaron Bertrand's new feedback we have come to a mutual conclusion

PARSENAME() is the inverse of QUOTENAME() if the following conditions are met:

PARSENAME(x,1)-1(QUOTENAME(x,q)) where q={1,2} and x='{'any_string_value'}



References:


PARSENAME (Transact SQL) (Microsoft Docs)
QUOTENAME (Transact SQL) (Microsoft Docs)