String[][] zoneStringsEn = DateTimeUtils.getDateFormatSymbols(Locale.ENGLISH).getZoneStrings();
for (String[] strings : zoneStringsEn) {
    if (strings != null && strings.length >= 5 && "America/Los_Angeles".equals(strings[0])) {
        System.out.println(strings[2]); // "PST" in my JVM
    }
}

public class MyNameProvider extends DefaultNameProvider {

    @Override
    public String getShortName(Locale locale, String id, String nameKey, boolean standardTime) {
        if (Locale.ENGLISH.equals(locale) && "America/Los_Angeles".equals(id)) {
            // return the correct name for Daylight Saving Time
            return standardTime ? "PST" : "PDT";
        }

        return super.getShortName(locale, id, nameKey, standardTime);
    }
}

DateTimeZone.setNameProvider(new MyNameProvider());

DateTime d = new DateTime().withZone(DateTimeZone.forID("America/Los_Angeles"));
System.out.println(d.toString("dd MMM YYY HH:mm:ss z", Locale.ENGLISH));