Do both parents have to be cystic fibrosis carriers in order for their child to be born with this disease?

Yes: The gene for CF has been well characterized over the years and the delta 509 mutation represents the most prevalent. If both unaffected parents carry this mutation they have a relative risk of 1/4 of creating a baby with CF with any conception.(2/4 carrier, 1/4 nl) there are other CF mutations & of parents carry a different mutation (one 509 & one another) the intensity of symptoms can vary.

Yes: Both parents have to be carriers and then their child has 25% chance of having the disease, 50% chance of being a carrier and 25% of not even being a carrier. The most common mutation causing cystic fibrosis is Δf508. But there are over 1500 mutations identified so far.