Actually, they don't, or haven't. This was a case of a child with a pre-existing serious condition that would have surfaced anyway (mitochondrial abnormalities), and which had a premature onset due to being given an injection of 9 vaccines at one time. She became seriously ill, and receded into the pre-existing condition.
Autism is not caused by vaccinations, but sick children may experience the onset of illnesses prematurely due to being vaccinated with 9 viruses at the same time. Vaccinations are safe, and should be given to all healthy children. It's extremely ... off the chart rare ... for a child to have a bad reaction to vaccinations. Vaccinations are, after all, just a dead or live virus of a disease that was almost eradicated. The vaccinations have made the world a safer place, and healthy children should continue to be vaccinated.
Parents should also have a look at the doctor that started the rumour about mercury, or vaccinations, causing autism. He is no longer allowed to practice in Britain and the US. His paper has been removed from Nature magazine, and there is no verification of his theory, which was not peer reviewed prior to first publicatoin. Peer review resulted in the article being removed. He was removed from practicing medicine in Britain, so he moved to the States; somewhere like Arizona. There he set up another practice and was again removed from practicing medicine. Was it the vaccine, or the mercury? Seems lawyers have tried to argue both when trying to get American families a hefty settlement for having a disabled child.