There's no easy way around FIOS's router rental. You basically need to use some Verizon router at some point since they use MOCA.
On my last bill/upgrade I saw I could buy a Quantum Actiontec instead of renting.
https://www.amazon.com/Verizon-FiOS-Quantum-Gateway-AC1750/dp/B00QRJ8YME I don't know what parental controls are like on this router.
You could just use your existing router as a bridge to a new third party router. But that's more equipment to go wrong/debug. I did that with the old FIOS router I had that didn't support 802.11n.