ASA decided after receiving 67 complaints about the ad, including letters from the International Meat Trade Association (IMTA), The National Farmers' Union (NFU), the Guild of Welsh Lamb and Beef Suppliers, and members of the public that it did not breach its code.
In a statement from ASA, it said that the claim made about child abuse in the ad was likely to be seen as a demonstration of the very fervent beliefs of PETA and was, therefore, unlikley to cause serious or widespread offence to those affected by child abuse.
A spokesman for the ASA said: "While the ASA appreciated that the ad carried an 'anti-meat' message and was intended to promote a meat-free lifestyle, we considered that parents were likely to realise that, if a food was withdrawn from a child's diet, the nutrients that food provided should be replaced.
"We considered that parents were unlikely to withdraw meat from their children's diet without having an alternative source for its nutrients as a result of seeing the ad."
The spokesman for ASA said that the ad was unlikely to mislead consumers into believing meat-eating was responsible for obesity because they considered that it was generally understood by consumers that meat-eating, in the context of a balanced diet was not the primary cause of obesity.
The NFU said the ASA made a wrong and perverse decision in failing to uphold complaints about the advertising campaign.
A spokesman for the NFU said: "The implication of this ruling is that any extremist organisation can say what it likes because the fact that it is extremist means its material will not be taken seriously...The fact of the matter is there is no ground for arguing that eating meat does, of itself, contribute to childhood obesity and for PETA to claim otherwise is both wrong and potentially damaging to health."
PETA campaign coordinator, Anita Singh, said: "We're very pleased the ASA agrees that we may express our view that feeding kids meat is child abuse.
"Even though the truth about breaking harmful eating habits may be hard to swallow, people have a right to hear it."
The NFU spokesman added that they would be lodging a request for an independent review at the earliest opportuninty.