Child - parent contraints in MYsql?

There is no such thing as a "conditional" constraint: Either your parent field allways references another row (i.e. no 0 allowed), or it is no constraint at all.

This is quite a normal usecase, typically you would work around it with an ON DELETE TRIGGER on the parent, that also deletes the children, selecting them by parent.

In your example you might consider something like

CREATE TRIGGER cascade_delete_children 
    BEFORE DELETE ON node 
    FOR EACH ROW 
         DELETE FROM node WHERE parent=OLD.id;

You can use NULL instead of 0 and define the FOREIGN KEY with cascading deletes:

parent INT UNSIGNED NULL DEFAULT NULL,    --- here is the parent term id
                                          --- if it's NULL the term has no parent
...

CONSTRAINT parent_fk FOREIGN KEY(parent)
    REFERENCES thisTable(id) 
        ON DELETE CASCADE 
        ON UPDATE RESTRICT