What are some Parallelisms for... 1 Gollum looking for the ring. 2 Thoring looking for the Arkenstone. 3 Smaug looking for the cup that Bilbo steal?
I'm sure that a number of people here are eager to answer your question, but they -- including me! -- are probably not sure what you are talking about when you write "parallelisms." Could you define that term or give an example?
If you mean what are some ways to describe how each of these characters desires the item they seek, I may be able to offer some help with at least the first item. Gollum is absolutely obsessed with the ring and wants nothing more than to be reunited with it. Whereas Thorin might be acting out of a sense of honor and Smaug out of a sense of insult (who should dare steal from him!), Gollum seems to be acting out an extreme, self-destructive obsession.
There are all kinds of allegorical readings of Tolkien's fiction, some better than others (see the links below for two examples). When you write "parallelisms," maybe you mean these sorts of allegorical readings.
In any case, please consider editing your question to provide more information about what you're looking for.