I am afraid it is a sad day when a stay at home parent is denied a credit card. I realize the best way to get a credit card is to have your own personal income source. When this is not available the next criteria can be to look at the household income.
Many years ago I saw a television program where a woman was having trouble establishing her own personal credit. I knew early on that good credit was important. It surprises me that a stay at home parent would not be able to use the household income to obtain a credit card.
If you are not the main account holder of the credit card you will not be able to make certain changes to the credit card account but you will be able to make all the purchases and pay the bills. The main account holder needs to make the request for a higher credit limit or raise the question about lowering the interest rate.
I hope that household income is considered in offering credit cards but I doubt that a stay at home parent will have the same issues with obtaining debit cards. In the past many people have forgone using their credit card and opted to use their debit card instead. Maybe the wave of the future will be to utilize cash more often.