No, it's not a common phrase at all. In normal English usage, it seems rather strange and not a typical expression you would hear in everyday conversation or see in general writing.
I'm not sure what specific 'dads cock story' you are referring to. It could potentially be a story about a father's pet rooster. Maybe it's a story about how the dad takes care of the rooster, like feeding it, building a coop for it, and the funny or interesting things that happen around the rooster.
Well, here's one. There was a dad who had a little bantam cock. He named it Sunny. Every morning, Sunny would wake the whole family up with its loud crowing. The dad would then take Sunny out to the yard and watch it strut around. One day, Sunny got into a fight with a wild bird that had come into the yard. The dad had to step in and separate them. It was a funny little story the dad would tell his friends about his feisty little cock.
I don't have a pre - known 'dads big balls story'. It might be a very specific family - related story within a particular community or family. Maybe it's about a dad who has some large balls in a metaphorical sense, like being very brave or having big dreams.