I wired transfer about $20k to my father's account for family support (and the limit for gift tax is ~$14300 per person), in this case, can I claim that I'm actually supporting both my mother and father?
Are you actually supporting both your mother and father? Is the account joint? Is your mother aware of the gift and has control over the money same as your father? If you cannot answer (and provide support) yes to all these questions, then I doubt you could make such a claim.
If the fact that the bank account is only in your father's name is a mere technicality for whatever reason, and the money is in fact intended, controlled by and benefits both of your parents, then I believe you can.
See more details about gift tax in the IRS publication 950.
As long as you don't mind doing the extra tax paperwork, you don't need to pay any taxes, since the gift tax doesn't kick in until you've given someone several million dollars over a lifetime.