I was assuming a multi-person video chat, probably on Google Hangouts. Although I understand your dad's rule, RealRedCloud. Would your dad allow you to participate if your camera was turned off, or perhaps pointed at the wall? Maybe cover the camera with a sticker? Some way so that you can be a part, but your face is not shown (I assume that's what the rule is about.)
As for specific day and time, what works for the most people? The best time for me is in the evenings after work (Pacific Time Zone.) Perhaps at 7pm (1900). However, I am more flexible on the weekends. Saturdays?