Well I’m not familiar with every one of Quito’s parks, but I do know that Parque Carolina has public tennis courts. Someone else mentioned that there are clay courts very near Parque Carolina, but I’ve not seen them so I can’t vouch for that myself. Also, there is a tennis & golf club in Quito and you can look at their website here: http://www.qtgc.com/
Swissotel also has tennis courts and they offer a very affordable package to anyone who wants to become a member of their health club and use their fitness center, swimming pool, etc. Same goes for Marriott and Hotel Quito, although I don’t recall that these latter have tennis courts…