Just discovered that we underpaid by about 30k over the past three years.  We received big refunds all three years and may not have even been eligible at all for refunded taxes.  Paid the mortgage on parents' house and took off interest and taxes.  Apparently not eligible for those deductions.

What to do now?

Talk to an accountant ASAP and try to sort it out before the govt. comes after you.  

https://www.hhcpa.com/blogs/income-tax-accountants-cpa/home-interest-deduction-even-when-its-not-in-your-name/

There are instances where you can claim these deductions, even if you aren't on the loan.  If you are somehow legally responsible for the property, or pay for the insurance, or have an agreement with your parents that you will get the house when they die (i.e. some type of ownership).  I think technically you need to be added to the title so you can show ownership, but if there is an oral agreement, you might have an argument...in the short term.  Long term, you probably need some written contract between you and your parents, and then get added to the title.  If the IRS audits you, if you can show an agreement/contract with your parents, you can then "play dumb" about not being added to the title, and promise to do it.
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I used to do taxes a long time ago, but i'm not an expert at all.  I'd ask a CPA; I know a lot of people who pay property taxes for family members and deduct them, but if it's a lot of $, you may get red flagged and audited, so you want to be prepared.  If you have a high salary but your personal home has low interest/taxes, then your additional payments for your parents might not raise any flags, but hard to say.
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In a recent Tax Court Summary Opinion, it has been decided that a taxpayer can claim home interest deductions for making payments on a mortgage even though the mortgage was not legally owned by the taxpayer. In order for the taxpayer to claim the home interest deduction, there has to be an oral agreement granting the taxpayer an interest in the home in return for paying the mortgage and property expenses, along with the taxpayers’ name ultimately being added to the legal title. This will result in the taxpayer becoming an equitable owner of the property.

I would also discuss the moral implications of the situation with a priest.

I believe that there's some dispute about whether underpaying taxes is akin to theft.  One could make a strong case that the government does not have a legal/constitutional right to take parts of our incomes.  I believe this to be true.  Also, the fact that the government spends our money on myriad immoral purposes might justify withholding our taxes.  If it weren't for the risk of going to jail, I would withhold my taxes in principles due to government funding of abortion, the immoral activities of Israel, prosecuting unjust wars, various levels of corruption, etc.  But the risk of jeopardizing my family by ending up in jail deters me from doing so.  So that's where the accountant would come into play.  There are companies that can help you settle the past debt with the government for a much smaller amount than what is owed.

That would depend if the underpayment was deliberate or if it was an honest mistake.

Mistakes are still meant to be amended. If your brother lent you something and you forgot all about it, only remembering years later, would you not return it then rather than decide "eh, too late"?

Anyway, moral implications aside, you don't want to f*** with the government when it comes to taxes.

There is a difference between deliberately stealing and an honest accounting error.

There is a difference between deliberately stealing and an honest accounting error.

The latter becomes the former if you don't make every effort to correct the error once you become aware of it.

That is of course under the assumption that the government has the right to tax you in the first place, but sadly that's a bit of an irrelevant question because we don't really have much choice regardless. Pay the taxes or the government will come after you and your family. That's just the way things are.

give unto Ceasar what is Ceasar's and give unto God what is God's - Jesus

Does not apply if "Caesar" does not have a just claim on your money.  Otherwise, he's simply a robber.

Based on this article provided by anonymous https://www.hhcpa.com/blogs/income-tax-accountants-cpa/home-interest-deduction-even-when-its-not-in-your-name/, I realized that we actually can claim the taxes, because we have a verbal agreement to acquire partial ownership of the house based on payments to the principal, but not including taxes and interest.  We are working on getting a legal contract signed in order to protect us if the IRS ever does come after us.