Wow, I have no idea. A search online said they're non-toxic to cats, horses, and dogs. But parrots are much smaller, and also not mammals. You may want to call up your avian vet and ask them over the phone, that will give you the best answer.
I don't know for sure, but botanically, Gerberas belong to the same family as sunflowers, dandelions, daisies and milk thistle. They also belong to the same family as boneseed and fireweed and senecio (all of which are quite toxic). So, I wouldn't chance it until you've had a proper answer from a Dept. of Ag., Botany Dept. or horticulturist who can tell you for certain. For myself, there's just something about the shape of the leaves and the texture of the sap that says 'poison'... dunno why... just a feelin..g.
Anyway, if you do find out, it'd be good if you let us know!