So your father is a Chinese citizen? From what I know (you can double check with the consulate), he needs to go back to his hometown (wherever his Hukou is) to apply for this permit.
What my wife did in the past was mail the required documents back to a trusted friend in her hometown to apply for her parents. I think the first time though does require it to be done in person first. Just renewing, it is not required.
I have never heard of a China residency card. If you mean either his ID card or Hukou, then he should get replacements for both as soon as possible. They are both important documents for a mainland person.
Hukou is not for where you reside but where (in China) you are from and who is in your family. I think it translates into 'household registration'.
Since your father is elderly, you should check with the consulate to get a visa to go to China first to help deal with his documents issues (if he actually needs to go back to China).