Okie Dokie. Push up on the ceiling in a few spots and see if it moves up and down. Check close to the walls as the ceiling framing should be strongest there. If it moves easily, it won't hold your body weight without sagging. However, if the ceiling framing is strong you may be able to screw a couple of lag hooks directly into the framing members.
Get a couple of 3/8" diameter x 4" long lag hooks. They should have 1 3/4" of thread to get a good bite in the wood.
Now I'm guessing the parents are saying there's no way you are going to rip out any drywall. So you must pinpoint the framing members and drive those hooks in. If you have a studfinder, use it back and forth in the area that you want to place a hook. Most studfinders have a bias to one side, so flip it around and slide it back and forth again. Each time it signals a framing member, put a tiny pencil mark right when the signal starts and again when it goes out. If you are lucky, the pencil marks will agree, and the ceiling joist will be right between the marks. But don't drill yet.
Use a magnet to search for screws or nails that were used to hang the drywall. Run the magnet along the line of the framing member. Hopefully, the magnet finds nails or screws that confrim what the studfinder told you. Spend a lot of time on the prep, checking that you really are sure of the joist's location and strength.
Now I'm thinking that you would want to use two joists, one for each ring. The closer to the wall, the stronger, but you probably don't want to be so close that you can't do your workout. Anyway, time to drill. Use a 3/16" drill bit to drill a pilot hole for the hook. Try to center it in the framing member. You can use a smaller bit to drill through your pilot hole at a sharp angle to see if you are centered.
If your hole is centered, screw the hook in. If not drill a new hole and fill the old hole. The screw should go in until the threads disappear, then a couple more turns. I would worry if it goes in too easily, with a lot of snapping wood. If it is too difficult to get it in all the way, remove it and drill with a 1/4" drill bit and try again.
Good luck, however you decide to mount them.