I don't think that's correct (but I'm willing to be corrected by an expert).

I believe the immune response is mainly triggered by the spike protein. If the spike protein of the variant is different enough to cause a degree of vaccine escape, then the spike protein of the vaccine should be different enough to not get immediately cleared from the body.

This is based on what I've understood from listening to interviews with prof. Sarah Gilbert, head of the Oxford team.

The parent is referring to a valid concern that with repeated doses (like from a subsequent booster), the mild antigenicity of the vector virus itself can be a problem. The Oxford vaccine uses a Chimpanzee adenovirus, which can infect human cells but not replicate inside them. There after delivering the vaccine payload, over time the protein coat of the vector will be chopped up and presented to the immune system, as are all cellular proteins. As you are unlikely to have been previously infected with a chimp adenovirus, it should elicit not much of an immune response, but the body will see it as foreign and start raising antibodies against the adenovirus proteins (adenoviruses have their own versions of "spike" proteins, among others). Of course, we would expect most of the immune response would be against the corona spike payload, but some fraction would inevitably be for the adenovirus as well.

When you get your second dose, you already have some antibodies against chimp adenovirus, which will potentially destroy some of the vector before it has a chance to be taken up by your cells. Third or fourth booster shots against coronavirus variants using the same chimp adenovirus vector would trigger worse reactions, destroying some portion of the vector before the payload can be delivered. The Russian Sputnik vaccine tries to reduce this likelihood by using different adenovirus vectors for each dose.

The good news is there are more than enough vectors to choose from.