Ah... sorry, I didn't look close enough. It looks like your web developer must have used the promotion slider and added some custom functionality. I say this because the promotion slider doesn't actually have a pause button, which is what threw me off initially.
There was code added to promo_slider.js in the plugin. Typically this is a bad practice since upgrading the plugin later would overwrite all of the changes.
Anyways, in the code, your developer simply forgot to take into account the case where a user pauses the slider and then clicks on another slide. It also looks like the pause functionality is only applied if the 'Pause slider on mouseover' option or shortcode attribute is set.
Basically, line 95 of the promo_slider.js file is where the issue occurs. It currently reads:
if(autoAdvance) sliderInterval = setInterval(function(){progress('forward', currentSlider, panelCount);}, (timeDelay * 1000));
It should be changed to something like this:
if(autoAdvance && playpause == 0) sliderInterval = setInterval(function(){progress('forward', currentSlider, panelCount);}, (timeDelay * 1000));