What is the correct folder structure for including local headers in my Arduino project?
The reason why you can't reference the parent folder of your sketch when compiling is because the Arduino IDE doesn't compile your sketch in that folder. Instead it writes out the files from the IDE itself into temporary files in a "build" folder and compiles them in there - so the parent folder then does not contain your header file because the IDE has no clue about that.

So you need to put your header file somewhere that the IDE does know about - and the simplest place is where the libraries are.

A library is nothing more than a header file in a folder (both named the same) with optional source and other header files.  If you place your header in a folder in your libraries folder the IDE will see it as a library and you can include it into any sketch.

So you would have:

Documents/
    Arduino/
        libraries/
            MyHeader/
                MyHeader.h
    MyFancyProject/
        NodeOne/
            NodeOne.ino
        NodeTwo/
            NodeTwo.ino
        NodeThree/
            NodeThree.ino