For the payment problem, try using PayPal. Note that in my experience it defaults to charging you in dollars rather than euros, which will probably cost you a bit more. There's a not-terribly-prominent place to change that.
I'll take a look at the Parc Guell website now and will post again if I can figure out what the problem is.
Updated to add: The Parc Guell website's ticket-purchase process seems to be working on my desktop computer with an old version of Internet Explorer. You have to indicate how many tickets you want (screen initially shows 0) before progressing to the next screen where you choose a date. I was able to choose a date and time. That screen is color-coded, I assume to indicate how many tickets remain available for each time-slot.
The separate ticket-availability page (outside the purchase process) is not working for me. It's after midnight in Spain; perhaps there some sort of website maintenance going on that is affecting the availability page.
If nothing above explains why you were having difficulty, try using a different browser. If still no luck, use a different type of device (smartphone or tablet if you were using desktop; desktop if you were using a smartphone or tablet). Those technique fairly frequently work.