Instead of Position field you can use a field as LastSongId where you can point the last SongId.
So when you want to change you order you can update only two rows.
The first row: Your song with new order
The second row / the third row: The song before and after your your changed position song
Your new schema:
+------+------------+----------+
|SongId| PlaylistId |LastSongId|
+------+------------+----------+
| 1 | 10 | NULL |
| 2 | 10 | 1 |
| 3 | 10 | 2 |
| 5 | 10 | 3 |
| 6 | 11 | NULL |
+------+------------+----------+
If you change the SongId 3 as position 1, you'll have:
+------+------------+----------+
|SongId| PlaylistId |LastSongId|
+------+------------+----------+
| 1 | 10 | 3 |
| 2 | 10 | 1 |
| 3 | 10 | NULL |
| 5 | 10 | 2 |
| 6 | 11 | NULL |
+------+------------+----------+