Yes and no. You get prions from eating a person that's also infected with prions. Basically if you eat cloned meat of yourself it should be fine as you either already have prions, or you don't already have prions. Prions manifest as either CJD if you got it naturally or Kuru if you got it through canibalism.
Eating brain/spinal cord is not required to contract a prion disease; prions can also be spread through any biological medium where protein material is located, for instance, blood. This is the reason why those who have CJD or other TSEs in their family are unable to donate blood.
The reason for that actually makes sense. It's rarely clear how and when someone contracted a prion disease at time of diagnosis, and often it is unclear which specific prion disease a person has. While it may seem that a father contracted a prion disease after you were born, it could also be that the father has an inheritable prion disease that you too may have inherited.