... from my trial and error it seems [the parser is] trying to do modulo using the return value of print function ( = "1" ) ...

Perl can tell you with perfect clarity (update: and lots of parentheses) exactly how it sees the syntax it's trying to compile. See core modules O and B.

Win8 Strawberry 5.8.9.5 (32)  Tue 07/04/2023 15:01:58
C:\@Work\Perl\monks
>perl  -wMstrict  -MO=Deparse,-p  -le  "print (5+6) + (6*7) + (7**8);"
print (...) interpreted as function at -e line 1.
Useless use of addition (+) in void context at -e line 1.
BEGIN { $^W = 1; }
BEGIN { $/ = "\n"; $\ = "\n"; }
use strict 'refs';
((print(11) + 42) + 5764801);
-e syntax OK

>perl  -wMstrict  -MO=Deparse,-p  -le  "print +(5+6) + (6*7) + (7**8);
+"
BEGIN { $^W = 1; }
BEGIN { $/ = "\n"; $\ = "\n"; }
use strict 'refs';
print(5764854);
-e syntax OK


Win8 Strawberry 5.30.3.1 (64)  Tue 07/04/2023 15:03:46
C:\@Work\Perl\monks
>perl  -wMstrict  -MO=Deparse,-p  -le  "print (5+6) + (6*7) + (7**8);"
print (...) interpreted as function at -e line 1.
Useless use of addition (+) in void context at -e line 1.
BEGIN { $^W = 1; }
BEGIN { $/ = "\n"; $\ = "\n"; }
use strict;
((print(11) + 42) + 5764801);
-e syntax OK

>perl  -wMstrict  -MO=Deparse,-p  -le  "print +(5+6) + (6*7) + (7**8);
+"
BEGIN { $^W = 1; }
BEGIN { $/ = "\n"; $\ = "\n"; }
use strict;
print(5764854);
-e syntax OK
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