“I showed him gratitude.”
I - subject
Showed - verb
Gratitude - direct object
Him - indirect object
Correct, yes?
How does one parse this sentence?
“I showed gratitude to him.”
“To him” is a prepositional phrase, but what role does it play in the sentence? Is “him” still an indirect object if the sentence is altered in this way?
The prepositional phrase is the indirect object. Him is the object of the preposition.
Yes, the indirect object is expressed as a prepositional phrase. More explanation here.
I STILL have a hard time parsing these sometimes and this is supposed to be my area of study. I don’t know why, but I always take an extra minute. When words get rearranged, I often forget to think of the prepositional phrase as a unit even though I recognize it as a phrase.
and then you start looking at things like:
“She is bringing up her kids Catholic.”
“She is bringing up her kids to be Catholic.”
“She is bringing her kids up the stairs.”
“She is raising Catholic kids.”
“She is bringing her Catholic kids up to to be bilingual.”
:o
lol. Thanks for the answers guys. 
I don’t think that’s right, despite what the link says. Unless there has been redefinition of “indirect object” in the past few decades (entirely plausible). While the meaning of the examples in the OP are the same, the parsing is not. “To him” becomes just a plain ole prepositional phrase, which is not an indirect object, but probably an adverb phrase showing where the book was given.
Nope. By your parsing of the sentence, prepositional phrase, yes, but one instead of, and functionally equivalent to, an indirect object. It is not a “where” phrase at all, except in the trivial sense that there is often physical action accompanying giving. In “the boy built his mother a birdhouse”/"…a birdhouse for his mother", no physical action towards the mother is even implied.
If subject and direct object describe agent and patient respectively – John is the one doing the hitting, the nail is the thing being hit – then the indirect object or its prepopsitioinal-phrase substitute is the beneficiary of the action.