I have the same problem trying to play Air Force reloaded.
I did some testing out and here are the results:
Parallel Space installed from google store > No problem signing in to google play but GG cant find the game...
Parallel Space installed from this site > Cant sign in to google play but GG finds the game...
Conclusion:
The version of Parallel Space this site provides doesn't work with google play anymore.
I tried reinstall, reboot, installing past versions, nothing seem to work. Anyone with games connected to google play it still works for?
Thank you