In a common way, the single Connection
to the database is implemented through Sockets
mechanism. Better to say, that your single connection
is some kind of pipe, which is using for concrete query which is executing one per time.
It is hard to imagine such optimal and high-perfomance code implementation, where though one socket multiple queries and their results are sent in parallel. Such implementation could cause the confusion with data-packets and at least their lost.
As karmanaut mentioned above in comments, you need to implement DB pool, i.e. pool of connections, which will be used for each statement concurrently.