SyntaxError: JSON.parse: unexpected character at line 1 column 1 of the JSON data

The fact the character is a < make me think you have a PHP error, have you tried echoing all errors.

Since I don't have your database, I'm going through your code trying to find errors, so far, I've updated your JS file

$("#register-form").submit(function (event) {

    var entrance = $(this).find('input[name="IsValid"]').val();
    var password = $(this).find('input[name="objPassword"]').val();
    var namesurname = $(this).find('input[name="objNameSurname"]').val();
    var email = $(this).find('input[name="objEmail"]').val();
    var gsm = $(this).find('input[name="objGsm"]').val();
    var adres = $(this).find('input[name="objAddress"]').val();
    var termsOk = $(this).find('input[name="objAcceptTerms"]').val();

    var formURL = $(this).attr("action");


    if (request) {
        request.abort(); // cancel if any process on pending
    }

    var postData = {
        "objAskGrant": entrance,
        "objPass": password,
        "objNameSurname": namesurname,
        "objEmail": email,
        "objGsm": parseInt(gsm),
        "objAdres": adres,
        "objTerms": termsOk
    };

    $.post(formURL,postData,function(data,status){
        console.log("Data: " + data + "nStatus: " + status);
    });

    event.preventDefault();
});

PHP Edit:

 if (isset($_POST)) {

    $fValid = clear($_POST['objAskGrant']);
    $fTerms = clear($_POST['objTerms']);

    if ($fValid) {
        $fPass = clear($_POST['objPass']);
        $fNameSurname = clear($_POST['objNameSurname']);
        $fMail = clear($_POST['objEmail']);
        $fGsm = clear(int($_POST['objGsm']));
        $fAddress = clear($_POST['objAdres']);
        $UserIpAddress = "hidden";
        $UserCityLocation = "hidden";
        $UserCountry = "hidden";

        $DateTime = new DateTime();
        $result = $date->format('d-m-Y-H:i:s');
        $krr = explode('-', $result);
        $resultDateTime = implode("", $krr);

        $data = array('error' => 'Yükleme S?ras?nda Hata Olu?tu');

        $kayit = "INSERT INTO tbl_Records(UserNameSurname, UserMail, UserGsm, UserAddress, DateAdded, UserIp, UserCityLocation, UserCountry, IsChecked, GivenPasscode) VALUES ('$fNameSurname', '$fMail', '$fGsm', '$fAddress', '$resultDateTime', '$UserIpAddress', '$UserCityLocation', '$UserCountry', '$fTerms', '$fPass')";
        $retval = mysql_query( $kayit, $conn ); // Update with you connection details
            if ($retval) {
                $data = array('success' => 'Register Completed', 'postData' => $_POST);
            }

        } // valid ends
    }echo json_encode($data);

If you're doing the $.parseJSON(data) in an ajax success handler Since you're doing the $.parseJSON(data) in an ajax success handler, the problem is almost certainly that jQuery has already parsed it for you. jQuery will look at the Content-Type of the response and, if it's application/json, it will parse it, and provide the parsed result to your success handler. The first thing that will happen if you pass that into $.parseJSON will be that it will get converted back to a string ("[object Object]", in your case), which $.parseJSON will then fail to parse.

Just use data as-is, it's already an object, thanks to the automatic parsing:

$(document).ready(function() {
    $('#branchAndSubjects').click(function() {
        $.post('/findBranchAndSubjects', {
            roll: roll,
            _token: "{{csrf_token()}}"
        }, function(data) {
            console.log(data.year);             // 3
            console.log(data.subjects.length);  // 4
            console.log(data.subjects[0].name); // Control Systems
        });
    });
});