Hide related field in Add New popup form

Hide related field in Add New popup form

I have two forms:   Client and Pet.  Clients can be related to multiple Pets; each Pet has one related Client. 

On the Client form, I have a Pets multiple lookup field with the ability to Add New pets.

On the Pet form there exists the bi-directional single lookup back to Client. This is good in case I decide to add a Pet through the Pet form and not through the Client form.

However, when I choose to add a pet using the Add New function on the lookup field in the Client form, the popup shows the bidirectional Client field. Of course, since I'm creating this Pet FROM the Client form, I don't want to see this field on the popup Add Pet window. (...and it needs to relate back to the current Client, which I think it does...on Submit, it just doesn't show that in the Client field.)

How can I hide the Client lookup field on the popup form but keep it on the regular form?

Thanks!