How to improve the time taken to establish connections to database when using 400+ connections?

Question

• I have a multithreaded C application that connects to Database(DBaaS to be specific) using MySQL APIs.
• It makes around 400+ connections using mysql_real_connect() function.
• It takes around 10 minutes to make all the 400+ connections.

Can anyone suggest me how can I reduce the time taken for connection establishment? How can I achieve parallelism for the connections establishment?