They're completely connected. The mom in PA2 (Kristi) is the sister of Katie, the girl in PA1. Katie and Micah even appear a bunch of times in PA2 and the title cards even explain when they appear that it's a few weeks before Micah's death, making this a prequel to PA1.
PA2 explains how the demon was originally after Kristi because it wanted her baby, and the stepdaughter uncovers some evidence that a great-grandmother may have made a deal with a demon years ago to give up a first-born male child, and there's a scene between Katie and Kristi where talk about "stuff with mom" that happened when they were kids.
At the end, Kristi's family exorcises the demon by sending it after Katie instead, basically setting up PA1. They thought that was the end of it, but in the last few minutes of PA2, which takes place the same night PA1 ends, a possessed Katie turns up at their house, kills Kristi and her husband, and takes the baby. So it got what it wanted.
PA3 opens with a few scenes between the adult Katie and Kristi, but then goes back to a bunch of VHS tapes from the 80s showing how the demon came after them back them, and the grandmother was involved, opening up this whole other plotline involving covens and witchcraft.
