Correlation, not causation, is the problem here.
Nothing "causes" autism, it is a natural state for people. It's not even an "illness", it's a combination of personality and behavioral states that we classify as a condition simply because it is outside of "social norms". In short: "it just happens"
The parents have heard that vaccination causes Autism, so immediately after the vaccination they start to look for it and notice some behaviours.
This completely misses the point that those behaviours were already there, unnoticed. That child was always going to show these behaviours, the parent just didn't notice them until prompted. The fact that the vaccines are usually given at around the age a child starts to develop their own individual personality traits only serves to exacerbate this: it's easy to notice newly developing traits. Again, they would have developed anyway, they just happen to be at around the same time.
Note also that Autism is a spectrum not a cross this line diagnosis. Everyone is on the spectrum, it's just that below a certain (rough) point, we stop declaring it as "Autism" and start classifying people as having personality traits. A calm, nerdy, introverted person with low social skills and excellent memory function, for example, is further "up" the scale and more likely to be considered as autistic (or, more likely, having Asbergers) than an outgoing, sociable person with little ability to concentrate (who may be tagged with "ADHD" instead)
Some of these things are personality traits, some are a result of "how autistic" they are, others are both. In essence, though, they are attempt to categories people by their behavior and personality: they do not correspond to a disease or illness.
If you look hard enough you can notice autistic-like behaviour in anybody. I have little ability to emphathise and fairly "tame" emotional response, for example: that doesn't make me Autistic, but it perhaps suggests I'm further "up" the spectrum than somebody more emotionally involved: but I am, for the most part, a well adjusted individual, socially and behaviourally.
In short, they're looking for those behaviours and associating them with the vaccine. The behaviours existed before, and are just part of human existence.