Yes, I guarantee you that it IS going to be difficult to convince that you have parasites. The difference is that, unlike your GP and most other doctors, an infectious disease doctor is supposed to have had some sort of training in this field should have heard of these things before and should not act as if they don't exist at all (as done by most GPs).
They also have more of a clue what tests to order.
A tip from me is NOT to try and "convince" as if you specifically know you have some exact species of parasite (because you don't know exactly), rather calmly state that you have improved with antihelmintics (I assume you have???) and describe symptoms and observations, and what you think it MIGHT be. They will then at least do some tests, but if they do mention psychiatrics at all then that is the time to be firm, not beforehand.
Might not be a good idea to make a big deal about the internet, but I don't think there is anything wrong with mentioning the internet, as most people use the internet to check any health symptoms these days (as does my infectious diseases doctor who has frequently Googled with me by her side!) If you do mention it you should, however, state that you have read "medical journals" to get across that you only read valid peer-reviewed medical literature (as you should be doing) and not a lot of the quackery out there in "alternative medicine". Do NOT mention Curezone (while there is some good info here, there is a lot more nonsense information on this site, with people regurgitating the quackery of " Hulda Clark e", herbs, "zappers", "candida" etc. so make sure NOT to mention any of these things).
If you fail with one private doctor you can always get another (as I mentioned above, I had one idiot private gastroenterologist telling me I had "IBS" without doing any tests at all and recommending BS "cognitive behavioural therapy"). The infectious diseases doctor might, however, tell you to see a gastroenterologist first as the latter will do scans and other tests to rule out other non- parasite things. You should also see a gastroenterologist as well to rule out many other possibilities, and they should also know a little (not a lot) about parasites, especially given that was the course of action referred by your GP. (though this was just the order I saw people, so might be different).
I have another appointment next week with infectious diseases, but to be honest with you, I'm not expecting much and am improving on my own and hate the stress of seeing all these people. Though, of course still best to be checked out by a health professional regardless.