This problem occurs after dark. Photo of backyard is during daylight.

Buy one of those cantilever umbrellas. You would probably have to remove the present umbrella in the centre of the table, but the one I'm thinking of can be strategically tilted right on it's side to block sun/light. Then can be moved back for overhead shade.

check it out here.

I think I would just get good light blocking window treatments. It's nice to have some light for lingering after dark out back and security lighting you aren't paying for.

So is this question relating to when you are outside, sitting on the patio or inside?

We have the same problem out in the middle of the forest. These lights can illuminate houses up to a hundred yards away. Could you talk them into installing dark sky fixtures? Probably not, but might be worth a try. It is a growing movement.

It's an issue sitting out after dark. Not terrible if sitting down on the stone patio due to the trees but sitting up on the deck feels like we have a view of an airport runway.

Maybe add some Edison bulb hanging string lights for effect and warmth

Airport runway, lol! I get it now. Without seeing the deck, is there a way you could hang outdoor curtains panels that you could pull closed to help cut the light? I've seen lots of DIY ideas for these lately. You could take them down during the colder months.

do you have photos of deck area.