They believe that by exposing the youngsters to the diseases they will build up their own resistance to the bugs.
At least 500 sets of parents are said to be linked by the informal network - and each is ready to attend "measles parties" where families can meet those infected with the illnesses.
However the government has warned parents against shunning the vaccination scheme.
Mother-of-two Lesley Dove, from Harrow, Middlesex, is refusing to have her children treated with the MMR vaccine.
However so far Jonathan, eight, and Aurora, five, have failed to contract any of the illnesses.
Ms Dove said: "It is a matter of choice and letting parents make their own decisions.
"I think vaccines are bad for the immune system and it is better to let the body develop its own immunity.
"I would not say that all children should do this because there may be some who have weak immune systems.
"But I will not be letting my own children have the MMR vaccine."
Parents' network
Ms Dove is in contact with about 500 sets of parents and she lets them know if anyone in their area has children who are suffering from measles, mumps or German measles.
The children are then allowed to play with each other at the "parties".
Ms Dove, 36, added: "This is the way it was done until the days of MMR vaccine.
"When I was at school we all just caught these illnesses and got over them and developed our own immunity to them."
Their move is the latest development in the continuing debate over the MMR vaccine.
A spokeswoman for the Department of Health said: "We are extremely concerned that any parent might put the health of their child deliberately at risk in this way.
"The diseases these vaccines protect against are potentially very serious.
"It is every child's right to be protected against these diseases and mmunisation is the safest way for parents to protect their children."
Single vaccines
Other parents are campaigning for the right to give their children separate injections for the three illnesses.
However the British Medical Association recently rejected the idea of making single vaccines available.
They warned that parents who did not have their children vaccinated were threatening the health of other children.
Uptake of the MMR jab has fallen since concerns were first raised in 1998 that it was linked to a rise in autism and bowel disorders.
Immunisation rates have fallen to below recommended World Health Organisation levels, promoting fears that the immunity of the whole population could be threatened.