The explanation I got when learning how to do qPCR was that you need run both replicate experiments and parallel experiments to get publishable data.
Replicates (or biological replicates) would be qPCR runs on independently collected samples. That does not count your control vs your experimental groups. So for my work I had to collect 3 sets of control tissue and 3 sets of experimental tissue from independent sets of animals (in my case 3 generations but for mammalian work I think three different animals would be ok).
Parallel does imply that you ran the the samples together in the qPCR machine. I think the standard is that for each sample in a particular replicate you need to run three qPCR reactions in parallel. So in total you need to run 9 wells of qPCR reactions for each test group.