While as almost everyone stated already - it's impossible using just DNS. As a workaround I would suggest trying NGINX (http://nginx.org/en/docs/http/request_processing.html).
TL;DR - In NGINX you can create multiple virtual servers that can redirect your request based on the server name.
Ex. http://first.my-server.com redirect to place A and http://second.my-server.com redirect to place B, while both share a single physical server.
