Best neighborhood for pets that doesn’t cost an arm and a leg?

I am looking for an apartment preferably in midtown for me and my two small dogs. I spend a lot of time in New York and I think I’d like to take the leap and get a sublease at least to start since my lease is coming up. I have two small dogs and want them in an area where there is some kind of somewhere to walk them and not just sad gray concrete. My budget is around 4k. Any suggestions?

Thank you!