You certainly don't need to know C++ to be good at Perl.

Having a true object-oriented language in your background might help, but C++ is nowhere near object-oriented enough to make any significant difference in that regard (especially if you learned C first and then moved to C++, which will lead you to write mostly procedural C++ code); and a functional language (e.g., Lisp) would help a lot more, IMO; and in any case if your goal is to be good at Perl no other language will help you more than Perl itself.

Anything more than a cursory knowledge of C is actively detrimental (at first), because it leads to bad habits. This can be overcome, simply by working with Perl more, but there's no advantage to it (from a Perl perspective) until you reach the point (years down the road) where you're ready to start digging into the perl internals.

If your goal is to learn Perl, then you should study Perl. Write code in Perl that does useful things. Hang out on Perlmonks and read more than you post. Go back over your old code that you wrote in Perl a few months ago and notice which kinds of things that you did make it easy to understand and modify and which kinds of things make it difficult. Study Perl, use Perl, learn Perl.