My dog and I are wondering: is this a biting midge, or another bug?

It seems to be a Yellow-banded polybia wasp (Polybia occidentalis), a species of Potter wasp.

my doubts with this conclusion is that the abdomen of the insect in your photo is quite large compared to other observations of the species. Also, Polybia occidentalis is a species that exists in South and Central America, not up north in New York, Perhaps it is able to exist there due to the warm summer weather though I am not entirely sure.

It might be another species of potter wasp. Potter wasps are able to sting but rarely do so to large organisms and their stings don't hurt much. They usually prey on caterpillars and beetle larvae and use those to feed their own. They create these nests that look like a ceramic pot. each of these "pots" contains one larva of the wasp (along with all the food the wasp has collected for their larva when it hatches).

here's a photo of the pots (wasps seem to be willing to create these pots on pretty much any solid surface including under leaves):

Wasps are not parasites, they are hunters of other insects, if my identification of this species is correct, then you and your dog are likely not going to suffer any direct harm from this species let alone parasitic behaviour.