You can't file HOH unless you pay more than half the total expenses of the household. You probably need to file an amended return to change to single, but your parents' tax return will not be affected. This mistake only comes back on you.
Now, there is a much longer answer having to do with "what is a household?" For example, suppose you have a 3 bedroom apartment and you rent one of the bedrooms to your best friend. Even though you live under the same roof, you and your child can be considered a separate "household" from your friend. It depends on things like do you eat together, play together, take joint trips, share food and other household expenses; or do you live separate lives and she just pays rent. If you are separate households, then you only have to consider if you pay more than half the expenses of your household, and you don't have to think about your friend's income and expenses.
So we can also ask, are you and your child a separate household from your parents? It depends on the circumstances. Maybe you really do live separate lives, and you have a separate pantry for your food, and your parents don't take care of your child when you are out, and you take separate vacations. This seems unlikely for grandparents, but it is at least possible that some people might live that way.
It also matters who pays the household expenses, not just who has more money. Suppose you pay most of the household expenses because your parents spend all their money on fancy cars, expensive wine, trips to the casino, or medical expenses (which are considered personal expenses and not household expenses). You might pay more than half the combined expenses even though they have more money. Or suppose your parents are retired on social security and you really do pay more than half the total expenses. Then HOH would be perfectly fine.
Household expenses are rent or mortgage, utilities, insurance for the home or renters insurance, repairs and maintenance, property taxes, and food eaten in the home. Clothing, travel, medical expenses, food eaten outside the home, and entertainment are considered personal expenses and not household expenses.
So the long answer is, it depends on facts and circumstances, and sometimes a person living with their parents and their child will qualify to be head of household, either because there are two separate household or because the person really does pay more than half the total expenses.
The short answer is, most of the time, someone in your situation will not pay more than half the household expenses and does not qualify.
But you have to analyze the facts and circumstances. If you decide you are allowed to file HOH under the rules, write down your reasons and any proof, and keep it for 3 years in case of audit.
