If your child is in gymnastics and is loving the sport so much and they're begging for doing flips at home and are begging for something other than coming into the gym. If you have a lot of space and good equipment in order to use at home would be your basic floor bar. A very low, high 10 foot balance beam, something called a panel mat which you can fold up or fold down for a soft landing and probably some, a small springboard.
At home I do not suggest a lot of high flying gymnastics and basic skills on the beam, just getting your footing on the beam and balance. The bar should only be used for front supports and little skills, nothing great and the panel mat can be used for multiple things. Cartwheels, handstands, bridge kick overs, some back walk-overs, and maybe some back hand springs. Ultimately you don't wanna be practicing your gymnastics so much at home as you do want to just kind of tighten up the skills that you already know how to do, but those are some good pieces of equipment to start with if you have the room. Also you wanna make sure it's always padded, you don't to put it on concrete, you don't wanna put it in the backyard. You want to make sure it's even flooring and nice and soft. But yes, the pieces of equipment would be the bar, a beam, the pelt mat, and probably the spring board.