Under current rules, it doesn't matter if the account is in your name or your child's name - in either case it would be treated as a parental asset. If the account were in the grandparents' name, then it would not be listed on the FAFSA, but distributions from the 529 would be treated as tax-free income to the child in the year they were made, so the general rule of thumb was to wait until the child was a junior in college before using grandparent 529 funds (when possible).
They've just passed an overhaul of the FAFSA rules - google "FAFSA simplification" to read up on them - so everything in the previous paragraph is really not relevant after this year.
As a practical matter, the biggest factor is whether your parents or you can get a state income tax deduction for any contributions to the account, and which state 529 plan is involved may affect that if you and your parents live in different states. So if your parents are going to make contributions and they get a state tax deduction and it has to be to a 529 in their state, then they might just want to keep it in their name while they do that. If you're going to be the ones making most of the contributions and your state's rules say it has to be a 529 in your state, then it might make sense to either open your own 529 account (kids can have multiple 529s - my kids did for a while) or roll your parents' account over into your state plan.
If your parents might pass away or if you and your partner were headed for divorce, then as an administrative thing it might matter if the accounts were in your parents name or your name.
Transferring - if you decide that's what you want to do - is just a matter of calling up the 529 plan and changing the custodianship. It's not hard, they'll have a form to fill out, and there's really no best/worst way to do it.
Distributions from 529s for qualified educational expenses won't have any impact on your taxes. Although if it's still around when your kid goes to college, you might consider paying for the first $4K of expenses out of pocket and claiming the American Opportunity Tax Credit, which currently would save you between $1,000 and $2,500 on your federal income taxes.