Random Video Timeline Start?

I like using the “Video” background for an Avia “Color Section” on my homepage. However, every time a user hits the homepage, they start seeing the same ‘beginning’ to a 2 min video.

Since its a link to a YouTube video, I tried inputting the “time” variable into the URL so that at least it might start in the middle of the video timeline instead (i.e. “#t=1m01s“) Sadly this doesnt seem to work and just breaks the video.

Is it possible allow users to put in different time-line time stamps for the videos? Or could there be a selection to start the video at a “random” point on the timeline each time the page loads?

Thanks!