If I owe parking tickets in Minnesota will the state take my state refund to pay them off?

[Hinder]

They have the ability to.  Whether or not they will is probably dependent on your situation.  Here's some more information:

From Minnesota's Department of Revenue site:

The Revenue Recapture Act authorizes the Minnesota Department of Revenue to intercept (or “recapture”) your tax refunds to pay debts you owe to other state and public agencies, including the Minnesota college and university system, certain local governments, or the IRS.

Revenue recaptures only apply to individual tax refunds or other payments. The law doesn’t apply to corporations or corporate tax returns. Debts subject to recapture include, but are not limited to: claims for child support, medical bills, student loans, traffic fines and federal taxes.

In addition to income or property tax refunds, the department may recapture lottery winnings of more than $600 and Sustainable Forest Incentive Act payments.

You must be notified before Revenue Recapture is used. The claimant agency must send you a written notice within five days after it notifies the Department of Revenue. You have 45 days after receiving the notice to request a hearing to contest a revenue recapture.