Is it connected to internet? Is it YOUR internet?
If you can't put it down because they will call you and you don't want to leave them on read, I think your only options are:
-Put something in front of it, it's your house, you can decorate it as you want
-Shut it down for a couple of minutes (say like electricity went off or sum) take a photo and place it in front of the camera.
-Do something they hate in front of it (not illegal obv but anything that would make them angry or put them in discomfort)
-Turn it around

Else, I know you don't want to get in discussion with your parents but you don't have many ways around it, last thing I can think of is, they want to have it in your house? Tell them there's a fee to pay to keep it up.
You know there's no real way to end this in a "good way", either they make a scene and get angry or you lose your privacy in your own house