How to run a Parse Live Query Server (Web Sockets) behind an AWS Load Balancer?

If it works locally - eg, going around the LB then I would guess that the LB is not configured to pass requests to the backend servers on that port. Your LB will need some new rules to route traffic to you backed hosts based on the destination port your service is listening on, see AWS for more help with this

http://docs.aws.amazon.com/ElasticLoadBalancing/latest/DeveloperGuide/elb-add-or-delete-listeners.html

it looks like websockets are not officially supported on AWS ELB though, so you might need use one of the current workarounds https://forums.aws.amazon.com/thread.jspa?messageID=589328