You asked

How can I be a good parent at 3am?

It is not uncommon for your child wake up and cry as soon as you and your partner fall into a deep sleep. It is not good to argue about who checks on the baby when this happens. Unless plans have already been made, it is best that both of you get involved. It may be a nappy that needs changing or a formula bottle that needs to be prepared or something more serious.
To ease the discomfort of having to wake up at odd intervals during the night, you can try taking turns being on duty for alternate nights or divide each night into shifts. It might be better if you and your partner sleep separately during the week so that only one of you has interrupted sleep. It may be necessary for one of you to reduce working hours so that lack of sleep is not such a big issue.
If you are able, you can get family members to help out with looking after older children during the day, allowing you to catch up on lost sleep. You can put aside non-essential household chores for a while. This will save you energy and time.
Showing that you care about both your child's needs and your lack of sleep shows that you are being a good parent. Since being awake almost all night really disrupts your normal schedule, you will find that your definition of “good” and “bad” parenting starts to change as the sleepless nights accumulate.

More questions

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