Sometimes you just can't convince them not to worry (not that they don't have a reason; as mentioned before, bike touring is not the safest pastime), and you've got to be prepared for that possibility. My advice would be to treat this topic in conversations naturally, as if you know what you're doing (of course, you should actually know what you're doing), because it's the atypical and the unknown that scares people usually. If they get an idea in their heads that it's like an expedition, with you doing inhuman feats of endurance, braving the elements and sleeping with one eye open, fearing bandits or something, then it will be hard to convince them otherwise. And trust me, it doesn't take much for them to start thinking that way. Even though I've got many thousands of miles under my belt, in all possible road and weather conditions, they still think that doing 60 miles in a day is something extraordinary. We've watched touring documentaries together, as well as pro cycling races, with the peloton doing double the distance I do in half the time, yet they still think that bike touring is hard and not at all enjoyable.
Ultimately, you have to strike a balance between easing their worries and making concessions regarding your tour. Knowing that they will never stop worrying, due to the emotional, rather than rational causes, you might as well enjoy your tour, instead of trying to ease their slightest concern. Just try to keep your ambition in check, because seeing your plan through is even more important than having a bombproof plan, IMO. If you return home earlier than planned, or missing various pieces of kit due to losing them, that would be a huge setback in the trust department. If you do a few successful multi-day tours that go as planned, you will have a much easier time with your folks in 2-3 years time. I have no such luxury, having cut short a few tours, so I should know.