MySQL replication process parallel with user process?
It depends.

Older versions of MySQL run one replication query at a time.  Newer versions can do some level of parallelism.

Meanwhile, queries against the Slave can run in parallel just the same as reads and writes can go in parallel on a single server.  If the write (from replication) and the read (from a user on the slave) do not touch the same rows of the same tables, then they can go in parallel.  But if a significant lock is taken out by one, the other must wait.  Or get into a deadlock.

It's not clear where your INSERT is coming from.  It should not be coming from the Slave, since Slaves should be readonly.  If it is coming from the replication stream, then what version of MySQL is running and how is it configured?