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"I'm trying to research which credit cards to sign up for to ensure that we can get there using mostly miles."

There are no credit cards that will ensure that a family of four will be able to fly CHI-PAR roundtrip in August for mostly miles, unless you can charge hundreds of thousands of dollars within the next year. Take a look at current award availability (and redemption rates) for this August at aa.com, delta.com, and united.com. True, you would not necessarily wait until July, 2016, to attempt to book award seats for August, 2016, but even so, it's good to get an idea of what things look like for this summer. And it's likely going to take you several months to accumulate miles needed for even lowest-tier redemption rates for four roundtrips.

If you can split your party into two groups of two and fly on different flights (and maybe even on different airlines), you might have a better chance of achieving your goal. Good luck!