Although the park itself is free to get into, payment is required for admission to the monumental core. My boyfriend and I both decided that we wanted to visit, and did so on the last day of our visit to Barcelona, after having visited the Sagrada Familia in the morning.
We arrived at the park at 2pm to find that tickets to the monumental core had sold out, and that the next available admission was at 6:30pm. It was extremely hot and we had planned and booked things for the evening, and so unfortunately we didn’t hang around and wait.
The free area of the park was lovely, it made for a wonderful afternoon’s stroll. It’s a real shame that this is the only activity that we didn’t book in advance, and would recommend doing so if you wish to visit. More