Is it fair if the stay-home mom is the one who has to stay up taking care of the baby at night because the father is the breadwinner and has to sleep to be able to work the following day?
There is very little that’s “fair” about being a parent.
Raising your child is the responsibility of both parents, but the reality of life will often mean that while the responsibility is equally born, the actual amount of work will likely not be.
In your case, it sounds like Dad does need sleep to keep his job and continue to provide income for the family.
Mom as the stay at home parent is also doing a very important full time job – raising the child. However, Mom *might* be able to get sleep during the day (when the baby sleeps) which Dad cannot do at his job.
Is it “fair” that the stay at home parent has to catch their sleep in the form of catnaps whenever the baby sleeps? No. However it is not an unreasonable way to divvy up household responsibilities.
Note, if Dad were the stay at home parent then he should wake up at night and catch up on sleep during daytime catnaps.
Perhaps staggering sleeping hours of the parents could help alleviate this problem. Mom could like 8pm – 3 am, Dad 12am – 7 am or something like that.
Good news on this specific sleep question though – your child will likely sleep through the night at around 6 months of age.