Your JSON string:
{"screens":"{ 0: '1220x1080', 1: '1620x1080', 2: '1920x1080' }", "views":[2, 2, 61]}
is incorrect, as "screens" is a single string value (i.e. it has the value "{ 0: '1220x1080', 1: '1620x1080', 2: '1920x1080' }"). You want to remove the quotes so that it is an object with 3 value pairs ie.
{"screens": { "0": '1220x1080', "1": '1620x1080', "2": '1920x1080' }, "views":[2, 2, 61]}
See the following code which I've put in a fiddle for an example of this working:
var values = {"screens": { 0: '1220x1080', 1: '1620x1080', 2: '1920x1080' }, "views":[2, 2, 61]};
$('#test').sparkline(values.views,
{
type: "pie",
height: "100%",
tooltipFormat: '{{offset:offset}} - {{value}}',
tooltipValueLookups:
{
'offset': values.screens
}
});