Reasons for this behavior
The reason why this happens only with your partner is quite simple: the dog is jealous or territorial.
Dogs have personalities, just like humans. The adult dog was raised by your partner, so it's no wonder he likes her a lot. He wants her all for herself and doesn't want her to share her affection with the puppy. And just like with a spoiled child, you can (and probably should) regulate this behavior with your reactions to it.
The puppy's reactions
To be honest, I wouldn't worry too much about the puppy's reaction. I would prefer the adult wouldn't be harsh enough to make her cry out, but we'll get to that in a second.
She will yelp and cry, lie on her back, and shrink away.
This is picture-perfect submissive behavior an great dog-to-dog communication. Puppies instinctively submit to all members of the pack until they reach puberty, and they are very theatrical about it. Even adult dogs often yelp when they submit, even though the dominant dog didn't actually hurt them.
What to do next?
You should tell your adult dog that this kind of behavior is not acceptable. If you don't comment on it, it could escalate to their situations as well.
- As soon as you see the adult dog coming to interrupt the puppies affection seeking, look him in the eyes and tell him "No!".
- If he doesn't stop, try keeping him physically away from the puppy with your outstretched arm or leg.
- If that still doesn't work, use your hand like a mouth (like a sock puppet without the sock) and "nip" the adult dog in the side of his neck or torso as soon as he starts being aggressive. Do not actually grab or hurt him. Just nudge your fingertips into his side.
All this establishes you as the pack leader, as the one who decides what kind of behavior is acceptable here. It also teaches your adult dog to turn down the aggression. I suspect that he'll still dominate the puppy, but that is perfectly normal and natural behavior.
It's very important that you correct unwanted behavior as soon as possible. When the adult dog has the puppy yelping and rolling under him is the latest possible moment for a correction. It would be better to read the body language and warn him off as soon as he shows aggressive body language like stiff posture, licking the front of his lips or holding the tail very high.
To aid you in understanding your dogs body language and mindset, please have a look at this Youtube video.