"Changing The Past" Again

If someone, seating himself on either a Wellsian Time Machine or an Andersonian Time Patrol timecycle, threatens to travel into the past and to prevent my parents from meeting, then departs/disappears, should I fear that I will cease to exist after his departure? If he was in the past for half an hour before preventing my parents from meeting, will I then cease to exist half an hour after his departure?

Of course, it is logically possible for me to cease to exist at any time but why should that happen after this guy's departure, let alone half an hour after? (Matter ceasing to exist would break conservation laws but not logical laws.)

If we are in a single timeline, then the fact that I exist now is sufficient proof either that he did not attempt to prevent my parents from meeting or that, if he tried, then he failed. If he succeeds in preventing them from meeting, then he initiates an alternative or divergent timeline in which I was never born and never existed, not an impossible sequence of events in which I was not born and did not grow up but nevertheless somehow managed first to exist before his departure and secondly to cease to exist after his departure! Nor, if any of that did somehow happen, would there be any reason for a half hour delay.

People think of different times as different places existing at the same time, thus that someone who departs from the twenty-first century and spends half an hour in the twentieth century is obliged to return to the twenty-first century half an hour after his departure from it because that was how long he had been away!

Regular blog readers will know that these arguments have been presented many times before but they always seem fresh and there is always someone who assumes that preventing someone's birth makes him disappear in the present.