Vue.js dropdown nested menu (keep parent open when child active)

You need to add a @click.stop on your nested elements to stop the event propagation:

<router-link :to="child.path" @click.stop>
   {{child.text}}
</router-link>

Here is the updated JSFiddle: https://jsfiddle.net/myeh0mvo/23/

You can learn more about event modifiers on this page of the documentation: https://vuejs.org/v2/guide/events.html#Event-Modifiers