Can I advertise products on social networking sites such as tumblr or facebook?

[Penguin♥]

That should be fine, it's your own personal space on that website so I don't see why not. You see fanpages for companies all the time, and I often see people advertising their Etsy shops and stuff on Tumblr. Both sites are pretty good for advertising, and it's not illegal at all, because it's on your own blog.

You might want to consider getting your dad his own website for advertising, just to make it easier. There are plenty of sites that would host one for free, or you could buy a domain (.com or something similar) pretty cheaply.

But until then, you'll be fine advertising, go for it!