MapMyRide does that with their premium MVP service. Runtastic Road Bike Pro does that with the combination of their premium app and free online service, with a paid service available as well. These services will show your route and ride stats along with your latest reported position.

Other services/apps will display your latest reported position. Apple's "Find My Friends" is one; Google's Latitude is another, but reportedly ending soon.

There are more of both types, some discussed recently in this forum.