We are exploring an option, where we will have aunlocked parsys in the editable template which allows certain components to be draggable by the page author. Page created by this template has the parsys where the user has the ability to drag and drop the components but however they can not add/edit the style system. Looks like it is not possible but wanted to confirm.
If we had added the component in the parsys in the editable template and configured the style system and unlocked it then the component is available in the resulting page with the styles, but that is not what is desired. We want author to configure or at least select the style system on the page only if the component is available.
Which version of AEM are you using? To add components inside responsivegrid or parsys, it should be unlocked which is correct, if you can drag and drop components inside parsys and component has design dialog with style system then you can apply styles.
Are you applying style to parsys or the components inside parsys? Make sure your parsys component is unlock.