I also assumed that these are his parents. (in my other comments) To be precise, doesn't need to be. Could be anybody's parents.
It is always good to read thoroughly. Thanks for posting. ;-)
Yes, you are right. It means, he generally is absorbed in thoughts and particularly thinks of his parents. Another way to say that in German. "Er denkt an seine Eltern."
In this particular sentence, the thoughts are the subject of the sentence (the 'doer' of the verb) and is thus in the Nominativ case.
In addition to that, genetiv articles don't work that way. To use the genitiv articles you have to have either two nouns, one belonging to the other (The thoughts of the boy/ the boy's thoughts) or a genitiv preposition. In that first case, the genitiv would only affect the owner, the boy in this case. The thoughts would remain in the nominative i.e. Die Gedanken des Junges
No. The sentence, by itself, does not have enough information in it to infer that his parents have died. What's more, in English if the sentence concerned his own parents it would be 'He thinks of his parents'; 'He thinks of the parents' implies that it is someone else's parents that he is thinking about.
Because bei means with in this context. Also, ideas being on someone doesn't make much sense. Thoughts being on someone is possible, meaning to think about them or contemplate them, but it's more objective and detached. Google is telling me that to think on something is "nachdenken". Thoughts being with somebody implies a sympathetic mindset towards them.
In my searches, I've only seen the prepositions 'bei' or 'über' with Gedanken. Prepositions are always difficult and it's important to remember that your new language doesn't worry about your actual language, you need to try and get into how the new language uses its prepositions, regardless of how it might be in your own.