Parioli is a rather big and elegant area and not too far from the center. It is well served by many bus lines and a taxi ride to the center would take 15-20 minutes (depends on traffic, time of the day.)
Rome city center is big, so finding a place that offers easy access (walking distance) to most tourist spot can be challenging.
If you want to stay in the Spanish Steps/Trevi area, check the Trevi district (“Rione Trevi”) or the Flaminio area.