If Spinal Parasite is a 1/1 because it has two +1/+1 counters on it, you can remove both counters to pay for its activated ability.

Any permanent is a legal target for Spinal Parasite’s ability, not just one with a counter on it. The player who controls Spinal Parasite chooses which counter to remove from the targeted permanent.

Yes, this card’s base power/toughness is -1/-1. If it has five +1/+1 counters on it, it’s a 4/4 creature.