My parent is giving me $25000. Toward a down payment on the purchase of a home. I am married however my name is the only name on the gift letter supplied to the lender because my name is the only one on the mortgage. Will my parent have to pay a gift tax? I have not received any other gifts from my parent.

If the US, the annual gift limit tax free is 14,000 for 2015. That's from each parent to each recipient, but you say it in the singular.
Your one parent could give you 14,000 and your spouse 11,000.

Just to emphasize the point - if yuor parents are married then the $25K gift can count as $12.5K from each, keeping thenm under the limit of teh annual gift tax exclusion. Or if you are narried the gift could be construed as $12.5K each to you and your wife. But even if the gift is truly from one parent to you alone, it is highly unlikely that any gift tax is actually owed, because there is a $5.43 million lifetime exclusion that your parent can take advantage of. Your parent must file a gift tax form with next year's income tax filing to document using up part of the lifetime exclusion, but no taxes are owed unless the lifetime exclusion is exhausted.