Most of the recreational zones, have camping areas. ie Las Lajas(just past Vilaflor). These also have the added bonus of having BBQ areas, running water and toilets. Although on most of the sites the toilet facilities close around 8pm. You can not camp in the National Park and the Rangers are very strict and it does entail fines.
Zonas de acampada - Official Website of Tenerife Tourism Corporation Above is a link that lists all recreational areas and camping zones and the facilities on them. They are mainly free but you do need to apply for a permit from the Cabildo at least 4 days before you go. You can contact them via the website. We didn't realise this to begin with and to be honest got away with it once, but the second time the Park Ranger did get abit stropy.
I would personally recommed the sites at Los Frailes and San Jose de los Llanos.
Hope you have a good time.