Most States allow you to kill them and grackles if they are depredating something or causing a nuisance. It might be in Federal Law. Most states adopt Federal Law on birds.
The dairy I shot had federal protection against the pigeons. Technically the dairy was in a county where no firearms are allowed to be discharged. this is a big reason the pigeons were so wildly out of control. Literally the pigeons around a major city would all fly to this dairy. Feds actually stepped in and said figure it out or we shut it down and wrote up paperwork right there stating the allowance of discharge of a firearm due to the unhealthy amount of pigeons on that dairy. I was in cuffs a few times, and at one point the whole neighborhood that bordered the dairy was lined with police and swat. The first few years was touchy for sure, but every time, I walked an officer down to the office and showed them the documents on the dairy. They made a few phone calls and it was wrapped up. It happened every few months, but after 3 years, I finally got a sit down with the chief of police for BCSO. I explained everything to him, and he was very fair. He explained how they cannot ignore a shots fired call. I totally understood that, but he said instead of sending out the platoon like they normally do, he would assign an officer to run lights an sirens to the dairy first and if I was there, he would touch base with me and let dispatch know. If I wasnt, he was going to send the rest. It was pretty cool, they assigned an old guy who would show up, wave at me, I would wave back and he would be on his way. Worked great for like 5 years, and then he must of retired because the next time it happened, two very young kids showed up, rolled to the back of the dairy, AR's drawn and faced out in the desert away from the dairy. I even leaned over to my partner and said " look at these bozos, we could sneak through the dairy cows and shoot both of them in the back they are so unaware of their surroundings". They just assumed it wasnt on the dairy. Needless to say, we had about 150 birds laid all over by that point as well. They were just clueless and full of adrenaline and not paying attention in the slightest. I left the gun with my buddy, weasled my way over slowly, and asked if I could help them. They said they were called to shots fired, and all I said is "yes, thats us" (WRONG THING TO SAY, but I didnt know, it had been so buttery smooth for years), and both AR's came pointed right at my chest. My other buddy flies off the handle, comes running through the cows screaming and yelling (he doesnt do good with that kind of stuff), those 3 are yelling like chicken heads and im kind of giggling thinking how dumb this all is over pigeon hunting. Now guns are drawn on my buddy acting a fool and lots of screaming and I finally just scream at the top of my lungs, EVERYONE CALM THE F*** DOWN! Taken back, everyone shuts up.... i tell the young officers, they are a bit out of line, explain the entire situation to them, and tell them they would be fools to arrest us but haul me off if they so wish. I tell them the history of the place and even tell them if they drive around the pen where the cruiser is, they will see over 100 pigeons on the ground. AR's are pointed down, they apologize, we all shake hands, I take one down to the office to see the documentation and that was it. I never saw another officer after that, but it wasnt long after that last incident when covid hit, dairy took a huge hit, and unfortunately the old man who owned it all took his life in the office from all the stress. That was the end of the dairy.