I agree with the above. It is generally the employee's responsibility to notify the benefits manager of the qualifying event has occurred, or will be occuring. In your case it is a limiting age by the dependent child.
Once you are offered continuation, you must elect to continue coverage within a certain time period, called the election period. If you don't tell the employer you want to continue coverage before the election period expires, you may lose your right to continue coverage. If you have the option of either the state or federal continuations, once you make your choice, you can't change your decision if the election period has expired.
You must pay the entire premium for the coverage, including the part your parent used to pay as well as the part the employer paid before the qualifying event. In addition, you may also be required to pay an administrative fee under certain circumstances for COBRA.
Please check into this early as noted by Ides and good luck. Take care and gentle hugs.