There are queer Palestinians. They are getting hit by bombs just as much as the cis-het ones. There are pro-lgbt Palestinians. They are severely oppressed but they exist.
You are broadbrushing an entire region and its people and conflating Hamas with Palestine. LGBT people who support Palestine (which the majority of them mean they oppose their treatment by Israel and Palestines right to statehood) is not the same as supporting an anti-lgbt group. Is saying that the people of Texas deserve democracy the same as supporting the Republican Party? Regions and the people of the region are not the same as individual groups.
What false equivalency is the cartoon making? No where does it say Palestine and Israel are the same. It is defending lgbt people for supporting Palestine. It is pointing out that lgbt Palestinians are also getting hit by bombs. That is literally it. It is also by a stalwart LGBT advocate.
I get why you are reacting negatively but you are adding context that is not there. You are one of the most passionate advocates for Equality for LGBT people of all stripes on this site. Surely you can see how this is only meant to divide our community.
The article in question is meant to divide and it is succeeding.