Does SQL Server execute queries in parallel?

You mean "concurrently". The answer is yes, with caveats that are too broad to discuss here. In fact, the whole point of RDBMS is concurrency.

"Parallel" has a precise meaning in SQL Server: "a single query is distributed over more than one processor core".

As long as your first query doesn't lock a table needed in your second query, they will run in parallel.

The queries run in parallel, as far as possible.

The database uses different locks for read and write, on rows, blocks or whole tables, depending on what you do.

If one query only reads from a table, another query can also read from the same table at the same time. If one query updates some records in a table, another query may still be able to read from the table as long as it doesn't read any records that were locked for the update.