People who are double jabbed are much less likely to contract it, and if they do, they're much less likely to transmit it. Therefore even if they hadn't known that they'd caught COVID, there's a much lower chance that they would be passing it on.
The Delta variant is very transmissible - did you see the case in Australia that was tracked to two people just walking past each other in a shop? If you're not vaccinated, you could very easily pick it up between your twice-weekly tests.
Also, I'm assuming you're doing the lateral flow tests - they're not especially reliable. This was written about the MHRA guidelines:
"Therefore, it has emphasised that lateral flow tests are only authorised to be used as a “red light” test in order to find infectious people and ensure they self-isolate quickly, and not as a “green light” for people who test negative to enjoy greater freedoms.
This is because, unlike PCR tests, lateral flow tests cannot detect very low levels of coronavirus in a sample. This means the test may not give a positive result if you have only recently been infected; are in the incubation period; or if you have mostly recovered."
So during your twice-weekly tests, even if they come back as negative, you could possibly be getting a false negative especially if you've only recently been infected. Presuming you go on to test positive on your next test and don't get another false negative, that's a minimum of a week you're spreading COVID around. And of course, that's assuming you've got a good technique in carrying out the test. If you're not collecting samples properly, the results will be even less reliable.
Having had COVID before provides virtually no protection at all against the variants currently circulating.I was reading a report earlier today which showed that natural immunity is more or less worthless against the Delta variant. A single jab offers weak protection but double jabs offers excellent protection.
So yes, I would feel much safer around your parents.
The wording of your post OP feels very much like you're trying to make a point about vaxxed vs unvaxed, and how the latter is safer but I have to say I firmly disagree with your logic. Twice weekly testing is giving you a false sense of security - and while you might be feeling brave about the risk of catching it, the same might not be the case for the people you spread it to and infect.