How can I anchor my threshold pergolas?

Use "Met- Posts" with Ground Spike to take uprights, available from DIY's

If you really want to anchor it because of a concern about movement or wind, attaching it to planters would only be of significant help if the planters themselves are filled with dirt (so they add a lot of weight) or they have strong bottoms to support dirt-filled pots (rather than simply being cosmetic surrounds for pots). If they were substantial, then you could simply put a bolt (galvanized) with a large washer through the side of each wooden planter and through each pergola upright. You could also consider designing the pergola so that it is a little wider than the patio so that you could simply anchor it in the ground (you could dig a hole by each post location, fill with concrete, and plant a U-bracket into the concrete).

Stick the bottom end in a decorative pot full of concrete. If you use a big pot, them just fill halfway, put some pieces of pvc all the way through the concrete (make drainage holes) you can still use the top part of the pot for plants. It will be heavy and pretty 😁

I have an 8x8 sunshelter that sits on the stone deck of my pool. I did not want to install it permanently as I did not want to drill into the stone, and because it's easy to dismantle, I put it away for the winter. The metal frame corners are L shaped in each corner. So I put a piece of wood across the frame at the bottom of the L and set a gallon jug of water in each corner.

This is my sunshelter and the weight of the 4 jugs of water works very well.