How do I cancel a subscription that was purchased as an app from a third party

Do you mean you bought it from the makers rather than through the Apple App Store? If so you will have to approach the people you bought it from to cancel the subscription.

If you bought it from the Apple App Store, even though it's an app not made by Apple, then you can cancel it easily - please see: How to cancel a subscription from Apple - Apple Support

See the following from How to cancel a subscription from Apple - Apple Support

If you don’t see Cancel, the subscription is already canceled and won't renew.

If you are paying for the subscription through Apple, see: How to cancel a subscription from Apple - Apple Support