Almost certainly you're using a standard file-based session. While your mail sending script is running, it keeps the session file locked, meaning that any other parallel requests to any other session-enabled scripts using that particular session ID will be locked out.
You need release the session lock by adding in a session_write_close()
call in the appropriate places (e.g. before any long-running code sections/scripts). That'll release the lock and allow other parallel requests to proceed.