Why we are not saving the parent pointer in "B+ Tree" for easy upward traversal in tree?

Will it affect much if I add a pointer to the parent Node to get simplicity during splitting and insertion process?

General Node would then look something like this :

class BPTreeNode{
    bool leaf;
    BPTreeNode *next;
    BPTreeNode *parent; //add-on
    std::vector < int* >pointers;
    std::vector < int >keys;
};

What are the challenges I might get in real life database system since right now.

I am only implementing it as a hobby project.

There are two reasons I can think of:

1. The algorithm for deleting a value from a B+tree may result in an internal block A that has too few child blocks. If neither the block at the left or right of A can pass an entry to A in order to resolve this violation, then block A needs to merge into a sibling block B. This means that all the child blocks of block A need to have their parent pointer updated to block B. This is additional work that increases (a lot) the number of blocks that need an update in a delete-operation.

2. It represents extra space that is really not needed for performing the standard B+Tree operations. When searching a value via a B+Tree you can easily keep track of the path to that leaf level and use it for backtracking upwards in the tree.