In MySQL können Sie die kleinste (oder größte) ID mit JOIN
behalten :
DELETE p
FROM pictures p JOIN
(SELECT p.picture_id, p.date, MIN(id) as min_id
FROM pictures p
WHERE p.picture_id = 2 AND p.date = '2018-07-13'
GROUP BY p.picture_id
) pp
ON p.picture_id = pp.picture_id AND p.date = pp.date AND p.id > p.min_id;