Monitoring the Postgres query parser

As I understand it, when you send a query to Postgres like:

SELECT id FROM table1 WHERE name = $1;

Postgres will create a query plan. The plan will be cached for the same query in the same session. But if you create a function:

CREATE OR REPLACE FUNCTION get_table1 (parname text) RETURNS bigint
LANGUAGE plpgsql AS 
$$BEGIN
  RETURN (SELECT id FROM table1 where name = parname);
END$$;

The query plan will be cached for all future sessions.

I'd like to verify this theory by checking how often the query analyzer is invoked. Is there a way to inspect the number of times Postgres parses a query?

If possible, I'd like to monitor things like #parses per second and the min/max/avg parse duration.