Duct tape is very heavy per square inch. I'd skip it. Coats & Clark Dual Duty thread is quite strong. With a wide, relatively close together zigzag stitch, it will hold. Turning the edges under is a given, then sew around the edge and finally a X from corner to corner of the patch. If you want a second row of stitching, don't go over the first - go inside at least 5/8 of an inch. Too many stitches will weaken the old nylon.
My white Ripstop is 2 yards minus a corner I used to patch a dog crate.
When I worked in a kindergarten, we play various games with a parachute, but I don't think any involved children ON the parachute. Around, under, but never on. Maybe the seams aren't strong enough for the pulling. Just a thought.