Their fears are based on reports - dismissed by health authorities as groundless - of an apparent link between the measles, mumps and rubella vaccine and autism.
The HPA has reported a 36% rise in cases of measles in 2008 compared with 2007, from 990 to 1,348.
Says HPA immunisation expert Dr Mary Ramsay said: "The year-on-year increase of measles across England and Wales is very worrying.
"Measles should not be taken lightly, as you can never tell who will go on to develop the more serious complications of pneumonia and encephalitis.
"This is why it's incredibly important to remember that measles isn't a harmless childhood disease, and that it is never too late to get your child immunised with the MMR vaccine."
Data shows 112 cases in babies under one, 265 in those aged between one and three-and-a-half, 432 in children aged from three years seven months to 11 and 286 in youngsters aged 12 to 18.
The World Health Organisation says 95% of children need to be vaccinated to ensure herd immunity.