At mid-night of the second day* the parent will wake up and turn on the lights, but the baby will remain sleeping in bed. The parent will just stand there for about 3 minutes, then jump up and disappear off the screen. (The lights will turn off again, automatically.) When the baby wakes up in the morning, s/he will be alone.
(* example: if your Tamagotchi character mated and had a baby at 10:30 AM Thursday, then the parent will leave the baby at midnight as Friday becomes Saturday.)
If your Tamagotchi is paused for some time after the baby is born, it takes longer for the parent to leave. If the clock is not allowed to naturally advance through midnight, the parent won't leave. (So, don't pause it for long periods of time, and don't adjust the clock forward / backward.)