What would imparsable input be? per the spec Invalid characters nick/server are NUL, CR, LF, SPACE, and chan type characters. NUL, CR, LF, and SPACE, are already handled by the main message regex, but I suppose since these are public api we should check for them as well. NUL isn't necessary as we don't have a length parameter. And chan type characters require parsing during connection which we can probably ignore for now.
So tl;dr I'll add some tests for CR, LF, and SPACE.

I guess something like if the nick has server characters in it, or vice versa. Maybe something with the separators reversed, e.g., nick@foo!user@server or nick!user@server!foo? I guess now that the regex is anchored, this sort of thing is less possible, but I'm not sure if they're outright barred?

Per rfc1459 § 2.3.1 a servername is an rfc952 hostname whcihc means ! and @ are invalid. The modern irc spec is a bit more strict as well.
rfc1459 also defines user as follows which matches with our nick handling.

   <user>       ::= <nonwhite> { <nonwhite> }
   <nonwhite>   ::= <any 8bit code except SPACE (0x20), NUL (0x0), CR
                     (0xd), and LF (0xa)>


I can't find anything about user/username in the modern ircv3 spec, so fall back to nick rules which would still make it impossible for the parser to fail (i think..)