The way the silhouettes move (turning heads etc.) tell me that there is no way this is a 2D sprite.
Why not? It could be. If you ever watch cartoons (whether Western cartoons or Japanese anime), or any of Disney's older non-3D movies like Snow White or Beauty and the Beast or The Lion King, that's all 2D. Why can't it be 2D? 
(yes, they intermix 3D in some cartoons and anime, but it's predominately 2D)
I think these are full 3d animated models simply without any light on them?
Quite possibly. That's a good guess that may prove correct.
If it was 3D, it'd likely be 3D prerendered into 2D sprites.
Same goes with the actual ground, trees and obstacles on the main plane?
No, that's all almost certainly 2D. Ofcourse it uses parallax scrolling, which is ridiculously easy to implement, even for beginners.
Try not to make the mistake that fancy term = complex. Many things are given fancy names, but can be as easy to implement as eupnea. 
Also, if anyone knows what game engine was this built in, that would be nice to know!
Why? Game engines don't dictate visual style. The art decides the art style. Some engines influence it slightly, but art style is 99% from the art and how it is used in-game.
They could've used any of a hundred different engines and came out with the exact same game.
For starters, you'll want to read the article: The Art Of Braid: Creating A Visual Identity For An Unusual Game - it'll get you 90% of the way there, visually.
Also, many many many many 2D games don't use an "engine" in the way you are thinking of engine. What even is an engine? Is it a magical word you once heard somewhere that you've decided you desperately need? It sounds impressive, everyone talks about it, but it's as vague as vapor. Do you know how to program? Then program a game! What's the difference between programming a game and using an engine? What's the difference between an engine and a map editor?
These are rhetorical questions, to get you to think about what engines are and are not; I'm not actually needing the answers. 
