As JC said, you can ride but not in the NP (illegal), the GR10 would be mostly hiking, I understand that for these 2 reasons, people cross the Pyrenees on the Spanish side more than on the French side.

Look at the link from JC :
http://pyrenees2005.free.fr/index.php?Les-etapes
I found this to be a very interesting route. If you need help with the translation, please ask ;-)