My parrot is biting me!?

Two weeks ago, I bought two lutino Indian Ringnecked Parrots. In the last couple of days, one has started biting me when it is out of the cage and playing on my dining table. The onset of the biting seemed to coincide with me introducing some parrot toys which both parrots seem to love. This one parrot gets over-excited, its pupils contract and dilate, and it dances around then bites my fingers whenever it sees them. It steps up happily (unless it doesn't want to go back into the cage), climbs onto me, will go to sleep or preen on my shoulder and will chatter with me and is starting to mimic me. My only problem is the biting when it is playing. The other parrot doesn't bite and loves to climb on me. The parrots don't have a problem with each other when they are both sitting on me but the biting, bossy one will try to take toys away from the more pive one. I have read about biting but most training tips relate to stepping up and my guys are fine with this. What can I do?