Adding to what Zenourn says...
zenourn: Although it is impossible to prove that WiFi doesn't increase the risk of cancer, the relative risk of brain cancer in humans given that you have been exposed to WiFi appears to be very close to 1.0 (i.e., at most only slightly increases/decreases cancer risk). Of course, given these results are based upon observational studies in humans and treatment studies in animals care is needed in interpretation. It isn't easy randomly assigning hundreds of thousands of children to control and 'treatment' groups where they receive different doses of WiFi radiation over several years and then compare their cancer incidence over the next 20 years.
On the one hand, children are very susceptible to contaminants in the environment (e.g. lead, alcohol, drugs are all worse for kids than aduts). Cell biology and cancer biology are very complicated, and children's immune systems are not fully developed. Is it not unreasonable to believe that non-ionising radiation has some effect on cell biology that we don't understand yet, when all we know is what we learn from these observational studies?
On the other hand, the chance of getting cancer is based on an aggregate of probabilities from various risks/carcinogens. Not even all smokers get lung cancer (only most of them). An oncologist won't tell you that risk factors x, y, and/or z caused your cancer - they just don't know. (Unless you hare a smoker.) Removing a small risk from the environment will only reduce the probability of getting cancer by that small amount.
What these parents need to realise is that brain cancer in children is relatively common, with an annual incidence around 3 per 100,000 children. Many children who develop cancer will have used WiFi devices but looking at all the cases of children who use WiFi and don't develop cancer shows that they are likely independent events. The combination of genes that the child gets from the parents plays a far greater role in the risk of the child developing brain cancer but it is much more satisfying to certain people to place blame on something life WiFi rather than the random combination of genes the child got from you.
The parents in the story believe that Wifi is an environmental factor that is within their control. I have two children, and I'd also like their risk of getting brain cancer to be as small as possible. However, the risk has to be (1) proven, (2) non-negligible and (3) not too inconvenient. All are subjective and need a judgement call (even (1) to a certain extent). Much better to stay healthy (strong immune system) and focus on the known, greater risks...
I wish people would focus their energy of things that actually have reasonable scientific evidence of influencing morbidity and mortality of children (vaccinations, sunburn, obesity, smoke exposure, prenatal alcohol and drug exposure, ...).
... though I wish you hadn't said vaccinations in that last sentence.
I have just noticed that there is a wireless AP about 1 metre from my son's head in bed at night, through the wall. I must go and move it.