Reply To: Child at (X0, Y0, Z0 2.5) not moving to parent (X-5, Y0, Z0) when I simply Ctrl P > Object.

First, lets take a look at what parenting is. It is changing the reference point in which transforms are calculated. So when you parent the cube the cube’s origin is use as the reference point instead of the world origin. That is why when you move the parent it moves the child. 

Second, Let’s take a look at Object and Keep Transform. If you are just parenting straight then They function the same. Now if you parent the cone to cube and then move the cube down 2.5 on Z axis. Then select the cone, shift+select the cube.001 and ctrl+P. you get 2 different results. Object just switches the reference point in which case the cone will jump. Keep Transform will apply the transforms(In this case the move downward) created by the parent-child relationship. Thus the cone doesn’t move. 

Third, what is the inverse table. This is a hidden matric that adjust the child in relation to the parent’s relation to the world origin. This one is kind of hard to explain, but easy to show. In your setup, select the cone, then shift+select the cube.001, and ctrl+P->object or keep transform. They both set the inverse table. Then with both still selected press Alt+G to clear the location transform. You will notice that the cone is moved down, but it is also moved +5 on the X axis even though all location transforms are 0. This is because cube.001 was -5 on the x location in reference to the world origin. The inverse(Opposite) of -5 is +5. Thus the cone has an inverse table with location of 5, 0, 0. 

Fourth, Keep transform without inverse and without inverse options. Without inverse is easy. It clears all transforms on the cone and changes it’s reference to the parent. Thus moving the cone’s origin to the cube.001’s origin. Transform without inverse does the effect you described. It applies the inverse to the child. So in this case if you look at the transform of the cone the x location now has a 5 in it.

Fifth, Get everything back the way you started. Cube at world, cube.001 at -5 X, and cone at 2.5 z. Select cone, shift+select cube.001, press ctrl+P->object/Keep transform, and press alt+p->clear parent inverse. This will remove the the +5x in the inverse table and not apply it. Thus moving the cone to above cube.001.

I hope this clears things up.