If I play with Dog 1 alone, like tug-o-war or something, Dog 2 gets jealous and starts grabbing Dog 1's ankles, biting his neck, trying to provoke him to play with her and not me.
How do I remedy this? I stopped bringing them to the park because she does it with other dogs as well. She's not being aggressive with him (and certainly never with me), but she really annoys him when we play together, and he ends up getting angry and snapping at her and they tumble around, etc etc.
I see it as perfectly normal behavior. Its asking a lot of a dog to have him/her sit around while there is play going on. Expecting a dog to do that is asking too much IMO. Figure out a way to play with both together.
I avoid this with my 4 dogs by providing each with one-on-one time with me, and, by keeping them separated at times, even if I'm not working with one of them. That way they tolerate "sitting in the wings," so to speak, while I'm working or playing with one of the other dogs.
This is an impulse control problem. The real issue here is dog 2 doesn't know what to do when you're playing with dog 1, so she invents a game. I would insist that she settles in on a mat while playing for a minute or two with dog 1, then rotate. But in order to get there from here both dogs need to learn how to settle in on a mat, and I would follow the protocol in our rev-up/cool-down sticky to achieve that. Working both dogs individually, and then generalize the behavior when one dog is on a mat.