This technically depends on your definition of "in debt". If you own a house and have a morgage on said house you are considered to be in debt because you still owe on your house. Same thing goes if you owe on your credit cards, student loans, car loan etc...
Just like buying a house, when a park goes out and builds a new multi-million dollar attraction, they go out and secure a loan to cover the cost of the new ride. Then they pay over X number of years on the loan. Parks don't just cut a check for the expense of an expansion or new ride, the cost of the new ride is spread out over a number of years (just like your house). All parks do this, even the family owned parks. Pat Koch has made comments before about being nervous about signing the loans for their recent Mammoth project because it represented the single most expensive ride they've ever built.
The majority of parks end up making money at the end of the year when all is said and done. For example, in 2011 Cedar Fair made $83,995 (in thousands of dollars) before paying taxes (for the year) but still owes money for its expansions. In 2010, they ended up losing money for the year.
An example of a company that hasn't made any money is Great Wolf. They haven't turned a profit on their business in the last 6 years, and is a major reason of why they were just sold. They were building their mega lodges and when the real estate bubble collapsed, they owed more money to the banks because of construction loans then the entire property was valued at.
So what I'm getting at is a lot of companies (not just amusement park companies) are actually "in debt," and owe money. Just as Gisco said above, the problem is when you can't make your loan payments.