How to store and reuse leaflet LatLngs in Database

Currently I save coordinates for an polygon in this format into database:

[52.32527, 12.99683],[52.32191, 13.76587],[52.45936, 13.79333],[52.68637, 13.39233],[52.69636, 13.03528]

Later I fetch an array of the polygons from database, so I have something like $polygon_array[$key]['coordinates'] where the latlngs are a string.

With PHP adding a polygon to map works:

But adding to map without PHP does not work neither throws an error:

 var polygon_array = JSON.parse( '' ); // transfer PHP Array to Javascript

 jQuery.each(polygon_array, function (index, value) {
  coords = polygon_array[index]['coordinates'];
  L.polygon([coords]).addTo(map);
 });

I think the problem is that Javascript handles the coords as string.

My Questions: How can I use entries from database for a valid latlngs parameter?
Isn't there an easy way to use the format above?