I usually fold manually when working with HTML files:
Then I use
zfat to fold and
za to toggle the fold.
at is a text object in this context. To see how this works you can use it in visual mode to see what will be affected:
If you want to use syntax folding you can set:
I find this a bit too much but it might suit you. To make it behave a bit better when dealing with multiline tags you could follow the advice given here.