> But the one who is spending whole day with a small kid and no one else is much more lonely

So...don't do that? Let the parent who works in the office come home and spend time with the kid, and go out for drinks (or hiking or the gym or whatever) with other friends. Do all the chores beforehand during the day, so that the working parent only has kid duty.

If both are working, both have chores and kid duty after work.