What alternatives exist when a table requires too many foreign keys?
If there's any way to group parts, you might be able to introduce intermediate tables as a workaround.  This won't work.

Parts
+ Table 1
+ Table 2
+ ...
+ Table 400


But something along these lines might.

Parts
+ RedOrangeYellow parts
  + Table 1
  + Table 2
  + ...
  + Table 200

+ GreenBlueIndigoViolet parts
  + Table 201
  + Table 202
  + ...
  + Table 400


I'd want to take a hard look at your DDL before I recommended doing this, though.  And if you do this, don't start throwing ID numbers all over the place. You ought to be able to join "Table 400" directly to "Parts" without including "GreenBlueIndigoViolet parts".