You do need to be license eligible for each series. You can find the schedule and what license you need if you log into members.iracing.com and go to Find Official Races...iRacing Race Schedule.
You can run a TEST, Practice or Hosted session with any car or track you have - you don't need to be license eligible to run those.
With the Street Stock series (both dirt and asphalt), we have two separate series.
One is for Rookies (where upper level licenses are ineligible).
The other is for upper level licenses (where Rookies are ineligible).
So if you had been running the Rookie Street Stock, and got promoted to D class, you now have to run the upper level one (iRacing Street Stock Class C).