What is this aircraft that has both piston and jet engines?
It is a double-hulled "YAK-110" (which is two Yak-55's), it was only recently completed and will be at Oskosh 2018 (and so will I, so I may be able to add additional pictures in August).

Here is an article about it. 


                
            

Here is another video with the jet engine at its first airshow (and the picture you have is a screenshot from it):


                
            

It also isn't the only aircraft set up this way (jet/piston), the Screamin Sasquatch is another airshow aircraft converted to have both.