A third of parents who help with their children's car insurance believe high premiums will force them to give up these payments after European gender equality rules come into force next month.
From December 21, insurers can no longer consider a customer's gender when calculating what their insurance premium or retirement income should be, following a ruling by the European Court of Justice last year.
Some 32% of parents surveyed who contribute to their child's insurance costs predicted that premiums will go up by such an extent that they will have to stop paying for their child's car insurance.
However, Aviva said these views may be premature, with industry predictions that premiums are likely to fall by up to 10% for young male drivers while increasing by up to a quarter for young women.
A person's sex has traditionally been taken into account when assessing insurance risk, with women typically benefiting from smaller car insurance premiums due to their lower accident rates.
Aviva found that there is still a large amount of confusion over the implications of the new gender rules, with only one in five parents saying they are confident about what the changes will mean.
Comparison website MoneySupermarket said it is yet to see car insurance providers applying vast pricing changes.
It said that early indications show the hikes are not as sharp as anticipated and are around 2% so far.
Clare Francis, consumer finance expert at MoneySupermarket.com, said: "For car insurance, consumers need to consider the full picture, and for that they need to think about all the details that affect the cost of cover. A raft of other factors, from postcode to profession and points on a licence, will influence the cost of insurance."