The animal ate the father of Jay, who was an engineer.
So who is the engineer here? Father or Jay? How can I use which, that, who to refer to the whole object or only to parts of the object?
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You can use a relative pronoun without ambiguity if you make sure it immediately follows its antecedent. If another noun comes between the pronoun and its antecedent, as it would in the sort of sentence you're proposing, you'll need to reword for clarity.
Try a similar sentence: "The designer asked to speak to the father of Jay, who was an engineer." That sentence seems to say that Jay was the engineer. If the father was the engineer, "who" needs to appear just after "father": "The designer asked to speak to Jay's father, who was an engineer."
The word-order, while it violates no rule, results in a logical ambiguity. As written, I'd take the sentence to mean that Jay, formerly an engineer, had a father who was, at some point in the past(either before or after Jay lost his engineering credential, was eaten by a lion. Your use of the past tense in your dependent clause also contributes to the confusion. Assuming Jay's dad was an engineer, "The lion ate Jay's father who was an engineer." Assuming Jay is an engineer, "The lion ate Jay's, the engineer's, father." Finally and just to cover all bases, "The lion, an engineer, ate Jay's father.