I can understand DH wanting to do it I am exactly the same and sounds like your parents were too and that may even feel stronger for him as it's more of a father son thing as well but if the money isn't there I'm not sure how you can consider it.
Maybe just go through the budget with him and remind him that there will only be one wage for six months.
Maybe if it can be worked in, offer to do the dollar for dollar thing when he finds a part time job and pray he is not a diligent saver lol.
I have four kids, three older boys and a toddler, over the years I have helped out the boys with various things and not always 'exactly' my middle son has had the most help as he has needed it the other boys have never even noticed or mentioned it.
The car age is tough!