Parse Json array without keys

What you're trying to parse, is a string.

{
    "status": "OK",
    "message": "this is your start and end coordinates",
    "data": "{\"start\":[\"35.778763\",\"51.427360\"],\"end\":[\"35.768779, 51.415002\"]}"
}

So it works like this:

//first, retrieve the data from the response JSON Object from the server
JSONObject response = new JSONObject(jsonString);
String status = response.getString("status");
String message = response.getString("message");
//Note this: "data" is a string as well, but we'll have to parse that later.
String data = response.getString("data");

//get the doubles from the arrays from the "data" component.
JSONObject dataObject = new JSONObject(data);
JSONArray start = dataObject.getJSONArray("start");
JSONArray end = dataObject.getJSONArray("end");

for (int i = 0; i < start.length(); i++) {
    String value = start.getString(i);
    //do something with the start String (parse to double first)
}
for (int i = 0; i < end.length(); i++) {
    String value = end.getString(i);
    //do something with end String (parse to double first)
}

So data is actually a String, but represents a JSONObject (which you'll have to parse), which, in its turn, contains two JSONArrays.

If data was a JSONObject instead of a String, the JSON would have looked like this:

{
    "status": "OK",
    "message": "this is your start and end coordinates",
    "data": {
        "start": [
            "35.778763",
            "51.427360"
        ],
        "end": [
            "35.768779", //note that in your example, this quote is missing (first quote on next line too)
            "51.415002"
        ]
    }
}