Parents can be a tricky issue when working with teens. They run the gamut from completely absent to helicopter parents. The trick is to figure out what kind of parent you are dealing with and address that issue individually.
These parents are incapable of leaving their "child" alone; they keep their offspring within sight at all times and often seem to overpower them to a certain extent. While this can be very frustrating for a YA Librarian you need to remember that you don't necessarily know the motivation behind the parent's actions. There could be a background reason why the parents feels they need to be present. There could be medical, legal or safety reasons. Your job is to provide the best YA programming/library experience possible - even if it means a parent is in the room.
Now, there are clingy parents who don't present much of a problem to the running of the event. I've got one mother who never leaves her daughter alone at a program, but will bring her laptop and set up in a corner of the room to work. She doesn't try to participate in the program and I sometimes actually forget she is present. There have also been times when an extra adult has come in handy and she's helped out.
Then there is the opposite. The parent who stays in the program and tries to be an active participant. What do you do then? The best approach I've found is to just talk to them. Explain to them that the program is for teens and having an adult try to actively participate has a negative impact on the teens' experience. Most of the time the parent is willing to move off to the side. You can also try distracting them by starting a conversation and letting them talk to you rather than pay attention to their teen (of course, this only works if you aren't having to provide a lot of instruction in your program).
The Extremely Conservative
If you work with teens chances are you've come into contact with the uber-conservative parent. They are the ones that won't allow their teen to attend any program featuring zombies or vampires, who ask for "clean" books for their teens and forbid their children from having anything to do with Harry Potter. You job is to provide for EVERY teen, this means dealing with both liberal and conservative extremes. Try to keep this in mind when you are planning programs and teen events. By "keep in mind" I don't mean cater to a specific family's needs, but make sure to offer some programming that ANY teen can attend. I've found the best way to deal with the conservative parents is to just talk to them. Once they know that you aren't out to subvert the values they are trying to instill in their child they are far more likely to allow their teens to attend most programs.
I find this group of parents to be the most difficult simply because I never see them! The Absent parent is one whose teen always shows up at the library by themselves and often for the majority of the day. One trick I've found for actually meeting the parents of my teens to to host several after-hours programs throughout the year. Teens who participate are required to be signed in and out by an adult and have a permission form, so I get to actually talk to a parent I might not otherwise see.
Truthfully, the best thing you can do as a YA Librarian is try to get to know the parents of your teens as much as possible. When the parents know you and know what you have to offer their teen they are far more likely to encourage their teen to participate in library programs.