They’re doing weird things with time. It’s easy to miss if you’re not really paying attention (and probably don’t work at all if you’re not watching in order).
Nate is the biological father.
Sometime in his past, Nate clearly said that he donated sperm to a sperm bank.
Before the Parker investigation started, Agent Edie had been about to be artificially inseminated. She changed her mind about the whole thing. And then started following the family around. (to new york & alaska). After she fell in love with Nate in Alaska (and he either didn’t notice and/or rejected her), and the case fell apart, she went back to her pre-Parker life.
She then decided she really wanted a baby, went to the sperm bank, started looking through photos and found Nate. The consultant tried to talk her out of that one (said something like “there are better choices”), but she insisted it was the one she wanted.
So - yes, the baby’s Nate’s, but he doesn’t have a strong reason to think that he is There have also been a few subtle (and not so subtle hints) that there’s unreliable narration going on - so, all of the flashbacks where they’re talking to their lawyer or any of the investigators are somewhat suspect.
That says, it’s funnier than it deserves to be.