What happened to Peter's parents?

In the film, this is slightly unclear. During a montage, Peter searches for information about Dr. Connors and his parents. This shows that Peter's parents died in a crash. However, later in the film there are two scenes, one where Peter yells at Uncle Ben, another where Dr. Connors is talking to Osborne's henchman, which both make it sound like their disappearance is a secret/mystery. I recall hearing dialogue in one of the trailers that also suggested their vanishing was a mystery.

I'm not the only one to notice this. This article posits that there were some last minute re-cuts made to the film. As a result, the released version contains hints that Peter's parents' fate is a mystery that Connors and Osborne are involved in.

The last two trailers released have Dr. Connors saying, mid-Lizard transformation, ‘If you want the truth about your parents, Peter, come and get it!’ What truth is that? There’s no ‘truth’ in the film, and Connors and Peter never have a good conversation about Peter’s parents. Going by the "ASM Deleted Scenes #1" image that line of dialogue may very well take place during the sewer encounter with Mr. Ratha.

At this point, we now know the correct answer; they died in a plane crash. This was fully established in The Amazing Spider-Man 2, where it is explicitly shown. There is also a DVD extra that shows Parker's father is alive, having survived the plane crash, but since it was a deleted scene it is not canon.

If you're looking for "out of movie universe" information, in The Amazing Spider-Man Annual #5 (published in 1968) it is revealed that The third Red Skull, Albert Malik, was responsible for their deaths when he had an assassin, the Finisher, kill the two by sabotaging their airplane and causing it to crash.