Your parents are buying a house for 187,500. they have a good credit rating, are making a 20% down payment , and expect to pay 1,575/month. the interest rate for the mortgage is 4.65%. what must their realized income be before each month?

I don't entirely understand the question. Let me do my best.
$187,500 is cost of house.
20%, or $37,500 is the down payment.
The loan amount would be $187,500 - $37,500 = $150,000.

If we assume the annual rate of the loan is 4.65%
Then the monthly rate would be 4.65%/12 = 0.3875%

If the loan is $150,000, the interest is 0.3875%
The interst for the first month is $150,000 * 0.3875% = $581.25.

You stated that their payment is $1,575.
So the amount that pays off the loan is $1,575 - $581.25 = $993.75.

At the end of the month, they owe $150,000 - $993.75 = $149,006.25

For the second month, the amount of the payment that goes towards interst is
$149,006.25 * 0.3875% = $577.40. and the amount that goes towards the loan is $997.60.

At the end of the second month they owe $148,008.65.

Regarding realized income, we recommend a monthly loan payment not to exceed 28% of the monthly income. So if a payment of $1,575 is 28% of Gross, then the math is : $1,575 = 0.28*Gross.
Gross = $5,625 monthly.
About $67,500 annually.
About $33.75 an hour.