How to convert a simple string to a DateTime object?

Your date string is not valid. Here is example of it:

String time = '2022-01-13';

DateTime parseDate = DateFormat("yyyy-MM-dd").parse(time); // to be date
var inputDate = DateTime.parse(parseDate.toString()); // to be string

var outputFormat = DateFormat('dd MMM yyyy'); // set format be for ex 13 Jan 2022
var outputDate = outputFormat.format(inputDate); // set to be a string

While it doesn't make much sense in this form, it will work:

import 'package:intl/intl.dart';

void main() {
  String date = '12';
  DateTime dateTime = DateFormat("MM").parse(date);
  print(dateTime);
}