If a parser can parse a non-deterministic grammar, is the parser non-deterministic?
No, it doesn't mean that.  A non-deterministic algorithm is one that uses non-determinism.  You're not using non-determinism; your algorithm is completely deterministic.

Here's what's tripped you up.  You can recognize a non-deterministic grammar using a deterministic algorithm.  Similarly, you can recognize whether a word is accepted by a non-deterministic finite automaton (NFA) using a deterministic algorithm (e.g., simply convert the NFA to a DFA, then check whether the DFA accepts the word).  So, no, just because the grammar is non-deterministic does not mean that we should call your parsing algorithm non-deterministic.