Bill and Judy are Leslie's parents. They're both writers, and they're kind of hippies in the same way that Miss Edmunds is. They have different values from most of the people in Lark Creek, and they're more interested in being themselves than worrying what other people think. Leslie calls them by their first names, which seems really weird to Jess. They seem to treat Leslie more like an equal and another adult than their child. We never hear of them have any problem with Leslie suddenly adopting a puppy, or going to church with her friend, or spending so much time out of the house unsupervised in general.
Judy's pretty absent-minded, especially when she's working on a book project. Bill's a bit more present – and Jess observes that Leslie, accordingly, loves being with her father – but he knows more about culture than, say, practical things. They operate at a level that seems pretty foreign to Jess, with conversation about "French politics or string quartets […] or how to save the timber wolves or redwoods or singing whales" (4.136). This is a language Leslie "speaks" too, and it's a big contrast to the total absence of conversation at Jess's home, which is usually filled with the sounds of television.
When Leslie dies, we learn more about her parents' grief from Bill. Bill is the one who thanks Jess for being Leslie's friend, who says he has to keep Leslie's dog, and who has to interact with the outside world. Their loss of Leslie seems to crush Bill and Judy. All their joy and excitement about being on the farm completely dissipates with Leslie's loss, and they just get out of town as fast as they can.