Jstree get all child nodes of root

Take a look at the real_checkboxes options. This might get you started:

        $("#jstree").jstree({
           "checkbox": {
              real_checkboxes: true,
              real_checkboxes_names: function (n) {
                 var nid = 0;
                 $(n).each(function (data) {
                    nid = $(this).attr("nodeid");
                 });
                 return (["check_" + nid, nid]);
              },
              two_state: true
           },
           "plugins": ["themes", "json_data", "ui", "checkbox"]
        })
        .bind('open_node.jstree', function (e, data) {
           $('#jstree').jstree('check_node', 'li[selected=selected]');
        })

You'll probably need to change the bind behaviour so that it checks the parent when a child is checked.

Use parents in jQuery to get all parents, filtering out by li because all tree items are li in jstree, try this:

var parents = data.rslt.obj.parents("li");

And for children use children in jQuery, like so:

var children = data.rslt.obj.parent().find('li');

EDIT Using the above, here's how to get all parent and children and put them in all an array for each:

Parents:

var parents = [];
data.rslt.obj.parents("li").each(function () {
    parents.push({ id: $(this).attr("id"), description: $(this).children("a").text() });
});

Children:

var children = [];
data.rslt.obj.find("li").each(function () {
    children.push({ id: $(this).attr("id"), description: $(this).children("a").text() });
});