What is the meaning of 零一年?
As you can see '零一天' means 'and a day' is because there is a '一年' before it.

一年零一天 = one year and a day

二千零一年 = two thousand and one years

二零零一年 = the year 2001


'零一年' is shot for '二零零一年' (the year 2001),
You cannot reduce '二千零一年'  to '零一年', but you can reduce '二零零一年' to '零一年'. Just like we can shorten 1997 (the year) to 97, but can't shorten 1,997 (the number) to 97