The new rule has just been announced. Starting from June 2nd 2015, Thailand Post asks everyone to show ID card when sending parcels to anywhere both in country and abroad at post office. The reason of this rule is to suppress the drug trade by the Thai government. If you are a Thai, you have to show your Thai ID card. If you are foreigners, you have to show your passport.
So, if you want to send a parcel at post office in Thailand, you have to take ID card or passport with you. If you cannot show the identity card, they will not send a parcel for you.