Carchas can incubate as well - have one player hit the Female Rhynio until she's below 10% health. During this, you ride the Carcha and feed it the Male Rhynio pheromone. When the Female is below 10% health it will automatically disengage from your friend (I assume if close enough) and start hitting the sacrifice tame. It'll have a really noticeable red cloud around the tame when the Female was successful, and then she'll fly off.

The incubation time will begin. You want a dino with a large health pool because it will start to go down as the incubation continues, and when the baby Rhynio hatches it will kill that tame. If the sacrifice tame dies before incubation is complete or if you cryo it, the incubation will terminate and progress will be lost.

I suggest having a 'box' to keep the tame sacrifice in while it incubates, because it takes some time and you want to keep the sacrifice tame safe and alive. I just did stone fence supports and then large stone walls to surround the tame. You can then cryo the baby Rhynio AFTER it hatches (explodes) from the sacrifice tame, bring it back to base, and raise it in safety.

Good luck!