I regularly ran around Parque Maria Luisa. The park is beautiful, but it can get old after a while. You will also probably enjoy running along the river and over the bridge to Parque del Alamillo. Running on the streets is not an option. You'll understand that very quickly upon your arrival. As for safety, it's no problem if you stay off of the streets.
I did a quick search and came up with Agrupacion de Deportistas de Sevilla at
www.adsevilla.com There is some info there, as well as links to more local running info. Good luck and have fun running in Sevilla!