In short, the answer is no.
The MySQL server is designed to simultaneously run parallel user sessions, but not to execute one given user session in several parts in parallel.
This is a personal opinion, but I would refrain from applying the optimization forward by making assumptions about how RDBMS works. It is better to first measure the request and see if the response time is really a real problem or not, and only then explore possible optimizations.
"Premature optimization is the root of all evil." (Donald Knut)
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