Best way to clean engine parts at home?

I use diesel fuel.

Since the cost of solvent went up I have been using diesel fuel. It is cheap and available and many gas stations. It is not flammable making it pretty safe for this use.
It breaks down grease quickly.
For minor jobs I use a product called purple power.   It works well for cleaning dirty engine parts.
For internal engine parts and the engine block internals I use Dawn Dish wash soap and warm water. Then dry and oil the parts quickly so they don't rust.
For the Tin, Oil Pan, Valve covers, Timing cover I drop them off at a local machine shop. He runs them through the hot tank.   There is sludge under the windage trays in some oil pans and valve covers that you can't get to.   The sludge drys out after the engine is apart for a few weeks as you overhaul and wait for parts.   When new oil is introduced to these sludge deposits they break up and can do damage to a new engine.
If your valve covers and oil pan do not have windage trays or areas that can't be cleaned just wash them out..