This is all lore compliant. The Githyanki lich-queen made a pact with Tiamat and now there's an alliance between red dragons and the Githyanki species. Red dragons, usually younger ones, allow Githyanki to ride them on their raids into the material world so they can accumulate experience and wealth, while Githyanki benefit in the form of having awesome dragon mounts.
There's usually a small degree of mutual respect between rider and dragon. The Githyanki do not own them like a human would a horse. It is more like a partnership.
