Can you automatically generate a parser for a type using type theory some how?
In programming languages, we usually distinguish concrete syntax, where a program is represented as a linear list of symbols, from abstract syntax, where programs have a tree structure.

The vast majority of PL theory, including most type systems, happens after parsing, working with abstract syntax.

So, it's impossible to generate a parser simply from type rules or definitions, because there are arbitrarily many concrete syntaxes that could correspond to any abstract syntax. The choice is a purely human one.

In some languages, like Haskell, the description of a data type can be used to generate parsers and printers for types, using the deriving mechanism of the typeclass system. But it chooses a syntax that corresponds to the Haskell one by convention, and they could have chosen to define it any other way.