My OBJ exporter [and the Pro one as far as I know] simply exports the geometry and groups/instances-geometry in the order they are found in the SKP's entities context - this in turn is probably affected by the order in which they were placed into the model itself. There can be no control over the OBJ file's ordering.
As an OBJ file is basically a text file you could manually edit it with Notepad++ and reorder the hundreds of lines of code individual 'g' definition, but it'd be so easy to break it I wouldn't try.
Also we don't know how your third-party tool imports the parts anyway. If they are not in the order they are in the SKP but say alphabetical then as Dave says rename them !
You issue really isn't with Sketchup, perhaps it's with OBJ export, perhaps it about editing an OBJ file, perhaps it's about your other app - which you haven't given many clues about either π
R