Both of the options provided would ignore any indexes on your table because they wrap the column in a function. This is called a non-SARGable condition. You can drastically increase the speed of this query if you have an index on this column and use this method instead:
DELETE FROM Table_programs WHERE post_date LIKE '2013%';
As James K mentioned, you will need to ensure your data is all in the same format for this to be reliable.
Of all the various solutions posted I would suggest you go with the solution posted by lazerath - for the reason he mentioned - i.e., if there is an index on post_date, his solution will be able to make use of that index. None of the others, including the brilliant solution I had posted would do that.