You can get the first of the month, by calculating the last_day of the month before and add one day. It is awkward, but I think it is better than formatting a date as string and use that for calculation.
select
*
from
yourtable t
where
/* Greater or equal to the start of last month */
t.date >= DATE_ADD(LAST_DAY(DATE_SUB(NOW(), INTERVAL 2 MONTH)), INTERVAL 1 DAY) and
/* Smaller or equal than one month ago */
t.date <= DATE_SUB(NOW(), INTERVAL 1 MONTH)