1) Using $.each() iterate the parsed JSON.
2) To get the week number use the below protype method
Date.prototype.getWeek = function () {
    var onejan = new Date(this.getFullYear(), 0, 1);
    return Math.ceil((((this - onejan) / 86400000) + onejan.getDay() + 1) / 7);
}
To use:
var todayWeekNo = new Date().getWeek();
3)To check whether they belong to same week use
var isEqual = (todayWeekNo == new Date(j.Date).getWeek());
4) If it is not equal, delete it using index.
Finally,
Date.prototype.getWeek = function () {
    var onejan = new Date(this.getFullYear(), 0, 1);
    return Math.ceil((((this - onejan) / 86400000) + onejan.getDay() + 1) / 7);
}
var arr = [{
    "StudentID": 5041,
        "Status": "Joshua picked up  from school [0]    at 12:41PM and reached home [0] at    12:43PM",
        "Date": "2013-11-20"
}, {
    "StudentID": 5042,
        "Status": "Joshua picked up  from school [0]    at 12:41PM and reached home [0] at    12:43PM",
        "Date": "2013-11-20"
}, {
    "StudentID": 5043,
        "Status": "Joshua picked up  from school [0]    at 12:41PM and reached home [0] at    12:43PM",
        "Date": "2013-11-20"
}];
var todayWeekNo = new Date().getWeek();
$.each(arr, function (i, j) {
    var isEqual = todayWeekNo == new Date(j.Date).getWeek();
    if (!isEqual) {
        delete arr[i];
    }
});
Updates:
Since  Delete won't remove the element from the array it will only set the
  element as undefined.
So I tried using arr.splice(i, 1); but it was not working.  With reference to this  question here is an alternative approach.
var todayWeekNo = new Date().getWeek();
for (var i = 0; i < arr.length;) {
    var isEqual = (todayWeekNo == new Date(arr[i].Date).getWeek());
    if (!isEqual) {
        arr.splice(i, 1);
    } else {
        i++;
    }
}
Hope you understand.