Almost half the parents saving for college are using 529 college savings plans, which have several tax advantages. Earnings grow tax free and distributions are tax-free so long as the funds are used for qualified educational expenses such as tuition and fees. In addition, some states allow tax-deductible contributions to 529 plans, even for plans based out of state.

Parents saving for college with a 529 plan reported an average balance of $32,000 saved, almost 50% more than parents saving without a 529 plan. Parents working with a financial advisor also reported saving more for college than those who didn’t have an advisor — $14,000 more, according to the Fidelity survey.

The firm itself said it has seen a 34% increase in openings for its 529 college savings plan accounts through the first half of 2017 compared to the same time frame last year. (Fidelity sponsors five 529 plans — four direct-sold and one sold through advisors.)

Many parents who have set up 529 plans, however, don’t have a clear understanding of plan fundamentals, said Hazel. Many don’t know that 529 savings can be used to pay for more than just tuition and fees — they can be used to pay for books and room and board, for example — or that a beneficiary can be changed at any time and investments within a plan can also be changed.

More important, many parents — 44% in the Fidelity survey — believe that savings, including those in a 529 plan, will significantly reduce the chances of their student receiving financial aid. The impact, however, is often minimal. Assets held in a 529 plan owned by a dependent student or his or her parents, for example, have a weighting of just 5.64% in the Expected Family Contribution calculation used in the Free Application for Federal Student Aid (FAFSA).

The survey found that parents expect to cover 51% of their children's college costs with savings and loans with the remainder split between scholarships and loans their children take out. Thirty-seven percent of parents of high schoolers and 68% of parents of preschoolers and younger children reported that they're still paying off their student loans.

The average family income of parents participating in the Fidelity survey is $103,000, including $118,000 average for those working with an advisor and $92,000 for those who aren't.