MapMyRide contains erroneous points on occasion
I have observed a similar problem with RunKeeper in the past (not on my iPhone 6 as of yet) and I believe it has to do with the phone struggling to accurately locate itself at that moment. iPhone's "zero-in" on their current location, starting with a big radius and shrinking it as it gets more accurate information. With that, it always assumes that it is in the center of the circle. When the phone chokes on its location MapMyRide will record that location even if it is erroneous, I believe. 

What I did with RunKeeper when that happened was simply edit the activity later on a computer. I removed the problematic point and the statistics and route returned to normal. I don't believe I almost ever had that happen more than one or two times in an activity and it occurred rarely at that. 

I have not used MapMyRide so I don't know if there is a way to edit your activity. But, in short, I would suspect the phone as the the app isn't the actual source of the location information - it just asks the computer for it.