about delete in BST

[abramhum.c.l]

Hi:
I feel confused about the delete in binary search tree.
The following is the code:

void delete(struct bst_node** node) {
    struct bst_node* old_node = *node;
    if ((*node)->left == NULL) {
        *node = (*node)->right;
        free_node(old_node);
    } else if ((*node)->right == NULL) {
        *node = (*node)->left;
        free_node(old_node);
    } else {
        delete node with two children
    }
}

This problem is from the current node and its parents. In the above codes, obviously,
I cannot image how the parent node connect to the current node's right child, if (*node)->left == NULL; In theoretically, to find out the parent node, I should write following:

struct bst_node* p;
if(current_node!=null)
{
p=current_node; current_node=current->right;
}

And then node p is the parent code of current. However, this algorithm not show in the function void delete(struct bst_node** node); why, and how to explain that.
Thanks a lot.