How do I convert the date from one format to another date object in another format without using any deprecated classes?

Use SimpleDateFormat#format:

DateFormat originalFormat = new SimpleDateFormat("MMMM dd, yyyy", Locale.ENGLISH);
DateFormat targetFormat = new SimpleDateFormat("yyyyMMdd");
Date date = originalFormat.parse("August 21, 2012");
String formattedDate = targetFormat.format(date);  // 20120821

Also note that parse takes a String, not a Date object, which is already parsed.