Error 1093 states that you can\'t UPDATE or DELETE using a subquery if your subquery queries the table you are deleting from.
So you can\'t do
delete
You can do
delete t1,t2
FROM table1 t1
INNER JOIN
table1 t2 ON (t2.something = t1.id);
For the query in the question, this should be equivalent:
delete A
from adjacencies A
join adjacencies B ON A.parent = B.parent AND B.child=@me AND B.parent != @me
join adjacencies C ON A.child = C.child AND C.parent=@me