The correct answer is "them"
Explanation:
In general terms, the object of a sentence is the entity affected by the action. Moreover, in sentences, it is possible to find indirect and direct objects. In the case of the direct object is the element that directly receives the action, while the indirect object describes the person that benefits from the action.
In the case of the sentence "The parts company will ship them the package" the direct object is "the package" as this is the object that is "send", and therefore the object directly related to the action. Besides this, the word "them" would be the indirect object because this is the beneficiary of the action and it is indirectly related to it.
