Exploring the free part of the park
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We only explored a small part of the park because we did not want to wait 5+ hours for the next scheduled ticket entrance - lesson learned, you should book tickets in advance if you want to see all of Gaudi's work up close. You can still see a decent amount from the free section of the park, but it's super busy and the upper levels are quite steep and hilly. We stuck to the flattest, most level ground to have a quick look around and get a good view of the city.
Transport & Parking
The park is situated up a fairly steep hill, but there is a car park with a few accessible spaces beside the park. We took the Metro and walked - it was quite a distance and there are steps/escalators to navigate if you don't drive all the way up to the top. You can also get a bus up the hill but I'm afraid I don't know the details.
As far as the park entrance and free section is concerned, I found it to be quite flat with wide (sandy/dirt track) paths and enough trees to dip into the shade if needed. What you can see is very limited here however, and we were so disappointed once we realised (at 10.30am) that the next available slot to explore the Monumental Zone would be 5.30pm! It looked like there were quite a lot of steps down to the Monumental Zone, but there is another entrance at a lower level which can be accessed by a parallel road (uphill) to the one we climbed. I believe this is the entrance beside the parking - or at least where I could see the parking - there were so many people!
There were toilets on the flat bit we were exploring, but I didn't use these so I unfortunately can't comment if they were accessible or not.
The staff we spoke to were informative and polite, even as they were breaking the bad news that we wouldn't be getting to see the Monumental Zone.
Anything else you wish to tell us?
Book tickets in advance, and be aware that a lot of this park is hilly or has steps to navigate.