We went around 5 PM too and got a spot on the road just down from Superica. You shouldn't have a problem.
We went last Saturday around 6:30 and there was still parking in the paid lot across the street. $6 at the terminal, $ or card. Enjoy!
Generally easy to find with the meters all around and 3 parking structures. One being a deck
There is a pay parking lot right across the street. At 5 pm you might be lucky enough to get parking there. It's about $6 for 8 hours, but that price might be different on a weekend. There is also on street metered parking (free after 6 pm weekdays and all weekends) but there is no guarantee those spaces will be available. Moreover, Barcelona doesn't take reservations so going at 5 pm is wise. Shrimps in garlic and oil and the salad with olives and tomatoes and hearts of romaine are excellent.
There is on-street parking as well as a paid lot across the street.
Some street parking, a pay lot across the street, and a garage about a block away.
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