The Hotel Camou, opposite the church, used to advertise safe storage for bicycles but I'm not sure for how long.
This is from the website of the albergue Esprit du Chemin and refers to aprking for cars, but perhaps you can come to some arrangement for your bike.
If you want to park your car for a longer period, to walk (a part of) the Camino from Saint-Jean-Pied-de-Port, there are three possibilities:
•a public parking place: this is free, but not guarded. The best option is, to park your car near the sports hall “Jaï-Alaï”. The pilgrims office’s volunteers, or we, can give you directions,
•a place at a camping, for example camping de l’Arradoy (Joseph Iribarne, tel. (0033) (0)5 59 37 11 75). This will cost about € 1,50 a day. You yourself will have to make the necessary arrangements,
•the Renault garage, Route de Bayonne, Uhart Cize (next to Saint-Jean-Pied-de-Port), tel. (0033) (0)5 59 37 00 57. This will probably cost about € 40 a week. Keep in mind that the garage will be closed on Saturday afternoon and on Sunday.