Yes both options are possible.

As long as you have a highspeed connection as you mentioned, you will be able to view and control a slingbox which is connected at your parents house.

This can be viewed from a computer. If your computer incorporates a video out put card which has a second output such as s-vhs or DVI or HDMI, hen you can connect that to the tv of your choice (taking for granted you tv has the appropriate matching connection.