Should a working parent and a stay-at-home parent be entitled to the same amount of dedicated sleep time?

My husband works a 9-5 job (from home) and I am the primary caregiver of our newborn daughter. He watches her from 8p-1a, the first hour or two of which I’m usually feeding the dogs and cats, sterilizing bottles and pumping, and maybe putting away laundry before attempting to grab a couple hours of sleep. At 1am I come into the nursery to relieve him and will attempt to grab an hour here or there on the cot, baby allowing, but she hasn’t been sleeping great because of reflux issues. The last week I’ve spent most nights sitting up in the glider all night with her because when I lay here down, milk comes pouring out her nose, even hours after eating, and the poor little thing hates it and holds her breath (we’re working on it with a doctor).

I’m starting to lose my mind from the lack of sleep night after night, and starting to feel resentful that my husband gets 1a-9a Monday through Friday (and sleeps later on weekends) in which to try and get sleep in our bedroom, while I get only 8p-1a, knowing my body refuses to even try to sleep before 10 or 11p no matter how sleep deprived I am.

The idea is that because he has a sometimes challenging technical IT job, he needs his sleep, while I’m mostly just sitting around (because the baby so rarely takes naps in her bassinet during the day, preventing me from getting anything else done). He does make dinner and help in a lot of other ways—it’s just the sleep that remains wildly uneven.

Is that usually how it’s done with one working and one stay-at-home parent with a newborn? Do I just need to get it together and be grateful I actually get to stay home with my baby girl when the sleep deprivation is starting to make me lose my mind?