True, but FK (in child table) must reference a key (in parent), and most databases won't let you create a key without the underlying index.
The other direction, however, is not a given: most DBs will let you create a FK on fields not covered by an index, so deleting or modifying a parent can benefit if you create such index explicitly, because it can check for the existence of children much faster (and avoid potentially locking the entire table). Again, the access pattern governs what indexes are needed: if you never delete/modify parent, you may not need an index on FK (unless you also have some queries which can use it, of course).
The other direction, however, is not a given: most DBs will let you create a FK on fields not covered by an index, so deleting or modifying a parent can benefit if you create such index explicitly, because it can check for the existence of children much faster (and avoid potentially locking the entire table). Again, the access pattern governs what indexes are needed: if you never delete/modify parent, you may not need an index on FK (unless you also have some queries which can use it, of course).