Looks like I misunderstood the relevant passage in Chapter 49--when Quirrell confirms that other snake Animagi can’t overhear them, he isn’t implying that you also have to be a Parselmouth, he’s implying that he can only understand Harry because Harry wills it.
Voldemort is the last known Parselmouth, so it would be highly suspicious for Quirrell to also be one.
Looks like I misunderstood the relevant passage in Chapter 49--when Quirrell confirms that other snake Animagi can’t overhear them, he isn’t implying that you also have to be a Parselmouth, he’s implying that he can only understand Harry because Harry wills it.