I don't know if you've found a solution for this yet, but I guess you could use a SSIS package to process your cube. In the package, you could use two Analysis Services Processing tasks. You could select only the Parts dimension in the first task, and in the second, you could select the other objects. This way, the Parts dimension gets processed first. Once this is done, you could create a SQL Server job to run this package on a periodic basis.

I don't know if this is the optimal way to achieve this, but this is the only solution I could think of. If anyone else has a better solution, do let us know.

Regards,
Guru