Hello
I installed paradigma template in my joomla 2.5.17, allvideos 4.5.0 and php5.4

When I use K2 and insert the youtube video, it works but when I use it in an article or  a module. The plugin does not work.

medcare.vantage-communications.com/

I tried other formats (flv and mp4) and it did not work also

Are Joomla! plugins enabled for these views?

To use the plugin in videos, the module should render content plugins.
Eg. in custom HTML modules make sure the Prepare Content setting is enabled.