No, if you are the licensee of a series (i.e., TEDxNewYorkCity), then only you have the option to expand that series by applying for additional TEDx license types using the TEDxNewYorkCity name (i.e., submit a salon application using the event name TEDxNewYorkCitySalon or a youth application under the name TEDxYouth@NewYorkCity). Your team member cannot apply to organize TEDxNewYorkCitySalon if you hold the license for TEDxNewYorkCity.
While you cannot hold two different location-based licenses at the same time, you, as the licensee, CAN hold two different license types, as long as the licenses fall under the same location-name series. However, please note that in order to organize additional TEDxNewYorkCity event types - salon, youth, etc. - you must apply for and be approved for each of these individual licenses.