It used to be true that visitors could have free access early in the morning but the city has now changed this timeframe to be accessible to residents only, now. See here.
We just visited it a couple days ago in the afternoon after Casa Batllo and La Pedrera. If I could do it over again I would only go at the first morning timeslot only. It was a madhouse full of people and that’s just not how I like to enjoy my outdoor/garden spaces. I just prioritized my early morning timeslot for Casa Batllo and I am happy with that decision; it was lovely and having early access to the house with few people was great. The parc was just a mess in comparison.
We went to the MNAC and it has nice views over the city as well.