Ok so right now you can't have a playbook that sets "async: True" as the default for every task in it (ditto for async_poll). I am totally fine for patches that allow this as a playbook option though and then it would set any tasks that didn't have it set.
We already do this for sudo/sudo_user/etc. However it's not true that the playbook itself would occur serially, you would be starting task 1, 2, 3, etc with no wait.
If you want the playbooks to all go parallel, that's different.
AWX is not going to offer this in the 1st release in a few weeks, but it's something I'm looking to build in a follow on, so allow jobs to spawn child jobs in parallel.
Alternatively you can do that now by a shell script that backgrounds some processes and waits on the PIDs, though the above is going to be a lot cleaner.