I would mention this:
"I will need at least a couple days per week to work from home to assist my wife."
because it sounds like a need you will have while employed. You want an understanding you will have this flexibility before agreeing (or learn now you will not have it and can make a more informed choice). You can include the reason if you want, but I think it's good professional practice to disclose any needs/expectations before accepting an offer.
Further, most employers understand people have obligations outside of their place of work. In fact, some even provide benefits to help with these obligations. If the offer letter does not state the benefits, you can ask for clarification (i.e., 'can I work from home?' 'is there any parental leave?' 'is there health insurance?', etc.)
Right after getting an offer is the time to negotiate, and I would use it to make sure you can assist your family without repercussions at work. Plus, if your PI doesn't understand this reasonable request, it helps to know now rather than shortly after starting a new contract.
(Congrats on the offer, by the way).