My wife's work associates threw a retirement party for her at a private club, but I would also like to acknowledge her accomplishments with a family gathering. Is it OK to invite some of the same people from her company that attend the work party?
Yes, it's perfectly fine. You may even wish to ask them to say a few words about your wife's achievements. As her office peers may have already contributed to a retirement gift, you might wish to emphasize that they don't need to bring a gift to the second celebration.
Do you have any suggestions for celebrating our father's retirement that won't make him feel too old?
Sure, you can plan a celebration around a hobby or an interest your father has. This indicates that with his retirement, he'll have time to focus on the finer things in life. An emphasis on the things that he enjoys will keep him feeling younger and excited for the burden-free years to come.