...lily beetles go on lilies....

.. and Rosemary Beetles eat Rosemary ...

The rosemary beetle (Chrysolina americana) can devour the leaves of rosemary, lavender, thyme, sage and some other related plants.

what plant is infested or is it a test question?

...or a parking problem!!

Lol. Volunteer...:o)
Which are long-stay beetles. ?

There's asparagus beetle on asparagus, a green mint beetle which lives on mint, and my favourite (I'm joking) the Colorado beetle on potatoes.
But don't forget a lot of beetles are beneficial, such as the ladybird or the ground beetles.
In general terms, if they move quickly about, they're good. If they stay still, they're worthy of squishing.