If parents are present in an adjacent room and taking a fitness class, do I need to obtain childcare licenses in order to have an adult that supervises and plays with them?

Right now, you are not a regular day care because the children are in the same proximity as the children.  To avoid any confusion, make sure you make it clear that this is not a day care, but instead, a 'supervised play room for the children of parents'.  Include a posting that even though a supervised play area is available for children, it is still the parents responsibility to supervise their own children.  This puts you more in the category of any place that offers a play are for children (McDonalds, Chuck E.Cheese, Chick fe le). 
With that being said, however, I would still encourage you to revisit the state's website for what 'qualifies' as a daycare provider.  Because of people operating daycare services out of their homes, the state consistently revisits the issue of what is and is not a child care... so you do what to stay on top of this issue.
You also need to visit with the company that provides premises liability for your business.  Children are at greater risk of getting injured than adults.  This means that you are more likely to get a claim for an injury associated with taking care of someone else's child.  You need to make sure that you have coverage for this function... overwise... your carrier could deny a claim on the basis that this was not the usual course of business.  Not all insurance providers require it... but most will and you need to know you are protected before you incur any potential liability.