Perhaps hiding the truth about Pete isn't ethical, but considering how famous the Millers are, also as dominant alphas, Pete's life would not be peaceful if he were to be openly recognized as Chase's son. Joshua also didn't know if they were going to keep contact after the contract ends, so it'd be stupid to tell him they have a son together.
I also think Joshua doesn't know Chase knows about Pete. Chase overheard the colleague bringing up Pete, Joshua never told him he's got a kid.

I truly love Joshua, and I don’t know if what I’m about to say will come across as controversial, but I have to admit that it never felt right to me that he didn’t tell Chase from the beginning that he was the father. I mean, Chase was literally driving himself crazy trying to figure out who he had slept with for the first time. He kept tormenting himself, thinking he might have assaulted someone while he was in rut. And then, I mean… that’s his son. He’s the father, and he doesn’t even know it. I love Chase, but I’m not going to defend that choice.#-.-)