Ah, figured it out. When I disabled parental control and then tried myresolver.com, I found out that the DNS server the mac was using was an openDNS server, no matter what address I used in the network settings. Then I remembered that I had installed DNSCrypt on my mac a long time ago. Once I disabled the DNSCrypt software, everything on the mac now behaves normally, even with the Parental Controls turned on.
Out of curiosity, when the router profile is set to “OpenDNS Family Shield”, there is an IP address listed: 208.67.222.123. But When I use myresolver.com, the DNS server that my computer uses is a different openDNS server (208.67.219.17 or 208.67.219.15). I presume this is fine since they are all openDNS servers, but I’m wondering if this is normal. Also, I noticed when I set the profile to “Google Public DNS”, the DNS server my computer uses goes to a completely difference address (74.125.80.65 or a comcast DNS server). Why isn’t this setting working as expected?