This suggestion may not help everyone due to differing family set-ups, however the ‘parenting team’ can take different forms. For example, the parents may not live together but could still be helping each other, or there may be a partner or close relative who is taking on a parental-type role. Whoever is in this important ‘team’, it’s worth thinking about how you can work together better.
Sleepless nights are a relatively short phase in your lives (even if it doesn’t feel like it!) and you can help each other through it. If one person is dealing with most of the night-wakings, e.g. for a breastfed child or a child who will only be settled by one parent, then that person should be offered regular opportunities to take a nap or at least some form of break.
The other parent could offer to look after an early-riser before they go to work, or take care of a baby who likes to stay up late in the evenings. Even a short cat-nap for a sleep-deprived parent will make a big difference.
The parent getting more sleep should also offer to take care of some additional chores on a regular basis, e.g. cooking or cleaning.