I'd say some of the more intelligent ones probably do. Saw a video of an elephant giving birth, and it was actually kinda interesting to see how the mother could tell as soon as the baby came out, and when she turned around she lifted her leg to ensure she does not step on the baby. It does not look like much, but have to consider the thought process that must have been going on. She was also kicking it lightly after, to clear the mucus from the passages. And moving it around with her trunk etc.
Some animals are perhaps not as careful/aware though. Sea horses are interesting, the mother lays eggs inside the male then the male is the one that gives birth and literally sprays thousands of tiny babies everywhere and they just get taken by the stream. "My job here is done" and the father and mother leaves and goes and does whatever it is sea horses do. Most of the babies die.