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Definitely think it's impossible with Perl, would love to be proven wrong though!

Perl 5 + -p0, 136 bytes

Similar to other answers, this removes all even bytes and leaves any odd bytes (which is truthy).

tr<�
 "$&(*,.02468:<>@BDFHJLNPRTVXZ\\^`bdfhjlnprtvxz|~€‚„†ˆŠŒŽ’”–˜šœž ¢¤¦¨ª¬®°²´¶¸º¼¾ÀÂÄÆÈÊÌÎÐÒÔÖØÚÜÞàâäæèêìîðòôöøúüþ><>d

Try it online!

Perl 5, 59 bytes

58 bytes code + 1 for -p0.

Returns empty string (falsy) if the contents are all even bytes and all odd bytes if not (truthy).

tr$ "\$&(*,.02468:<>\@BDFHJLNPRTVXZ\\^`bdfhjlnprtvxz|~$$d

Try it online!

Definitely think it's impossible with Perl, would love to be proven wrong though!

Perl 5, 57 bytes

56 bytes code + 1 for -p.

Returns empty string (falsy) if the contents are all even bytes and all odd bytes if not (truthy).

tr$"\$&(*,.02468:<>\@BDFHJLNPRTVXZ\\^`bdfhjlnprtvxz|~$$d

Try it online!

Perl 5, 59 bytes

58 bytes code + 1 for -p0.

Returns empty string (falsy) if the contents are all even bytes and all odd bytes if not (truthy).

tr$ "\$&(*,.02468:<>\@BDFHJLNPRTVXZ\\^`bdfhjlnprtvxz|~$$d

Try it online!