If you mean 'public transport' in the literal sense the answer is that it's possible but only just. The island's transport operator TITSA provide a great service to places people live but of course no one lives on Teide (that's one of the many things that make it amazing!). You could probably get up very early and change once or twice for the only service of the day. Details of routes on TITSA's website. Much easier would be to book a coach tour or rent a car. It may be Spain's highest mountain but the roads are fine. And car rental is cheap. Remember that although the National Park is an absolute must-see it's not a place to spend hours, unless you're a hardy hiker.