I always start off with the mildest solvent and work my way up. This often prevents damaging delicate things like plastics and paint, and I always test using a cotton swab with the solvent to test on a less visible section of the part to ensure no damage will occur.
Here's my typical order of attack:
- Soap and water (usually Simple Green)
- Denatured alcohol (alcohol stove fuel)
- Stoddard solvent (parts washing fluid)
- Paint thinner (turpentine)
- Lacquer thinner
- Methyl-ethyl-ketone (MEK)
I used to also include trichloroethane (methyl chloroform), in between alcohol and stoddard solvent, but you can't get it anymore. I still have one bottle, but I use it very, very rarely.
For your situation I'd use stoddard solvent... it won't harm the rubber or plastic bits, if any, and will thoroughly dissolve the grease. This is the brand I use in my parts washer:
https://www.tractorsupply.com/tsc/product/crown-psc-1000-parts-cleaner-5-gal