There appears to be only one example in the books of a non-natural-Parselmouth speaking Parseltongue. From Chapter of Deathly Hallows:
"But how did you get in there?" he asked, staring from the fangs to Ron. "You need to speak Parseltongue!"
"He did," whispered Hermione. "Show him, Ron!"
Ron made a horrible strangled hissing noise.
"It's what you did to open the locket," he told Harry apologetically. "I had to have a few goes to get it right, but," he shrugged modestly, "we got there in the end."
There are several interesting observations we can make from this
- Ron, though not a born Parselmouth, was capable of performing the Parseltongue sounds.
- Hermione apparently counts this as speaking Parseltongue.
- Harry hears Ron's sounds as a hissing noise rather than as words.
This is interesting, because the first two observations would seem to indicate that there is nothing special about Parseltongue. It is simply a language consisting of particular sounds, and it is possible for someone to learn how the sounds correspond to word meanings. The only difference, then, between a Parselmouth and a non-Parselmouth would be that the former is born knowing the language while the latter has to learn it.
The third observation, however, seems to indicate that there may have been something fundamentally different between the sounds Ron made and standard Parseltongue. Throughout the rest of the books, Harry does not realize that he is hearing (or speaking) Parseltongue rather than English until someone points it out to him, yet in this case he appears to have heard Ron's hissing noises in the same fashion as a non-Parselmouth would have. This would imply that there might be some difference between true Parseltongue and the hissing sounds that Ron made.
If that is indeed the case then we might suppose that while anyone can learn to match up particular hissing sounds to particular meanings, and thus speak to snakes and understand snakes, only a true Parselmouth would perceive the hissing sounds as perfect English (or whatever their native tongue is).
So unless there is some enchantment that prevents Muggles from memorizing sound=definition, which seems quite unlikely, it would appear that a Muggle could learn the language of snakes, albeit without being a true Parselmouth.
However, it is hard to state anything definitively here, because the books don't actually explain why Harry normally hears Parseltongue as English.
Further confusion is caused by the fact that Harry routinely has trouble speaking Parseltongue when not faced with a snake. This would imply that a Parselmouth doesn't exactly know Parseltongue; rather, somehow Parseltongue gets transformed into a regular language somewhere between the ears and the brain of the Parselmouth. So we might argue that not only can Muggles learn Parseltongue, they can even surpass natural Parselmouths.