Pairing Apple Watch on different carrier than iPhone on Verizon

I would like to have my dad to use my Apple Watch ( cell with T-Mobile) he has iPhone but his phone

plan on Verizon

I thought because I had a plan Apple Watch on T-Mobile that it should pair with his Apple ID and still use my plan when not near his phone ( for like emergency)

so question is can I set up watch as family on my plan

then use his Apple ID with his Verizon iphone and get messages and phone calls?