The city I lived twice and visited also twice. I lived for one mouth and half total.
Security is not good as Tokyo and Singapore. I will be afraid of running midnight alone.
I think running culture is rooted there. So, maybe it will be no problem during daytime.
Coast side, barceloneta seems good place for running. Monjuik hill seems also good. You can do hill training there.
I have never been to the park in Barcelona but possibly there will be big and comfortable park.
Trail running is very popular in this area. So, you can find good place for doing it in countryside.
Barcelona was ranked around 20th in the ranking by monocle. So, the point to concern is security. Aside from running, cycling is very popular in Spain. Some great places for biking must be there.