Holding Company : A holding company is a company that owns other companies‘ outstanding stock. The term usually refers to a company that does not produce goods or services itself; rather, its purpose is to own shares of other companies to form a corporate group. https://en.wikipedia.org/wiki/Holding_company
Parent Company: A parent company is a company that owns enough voting stock in another firm to control management and operation by doing and influencing or electing its board of directors. The second company is deemed a subsidiary of the parent company. https://en.wikipedia.org/wiki/Parent_company
Subsidiaries: a company controlled by a holding company.
There are little legal differences between the two types of company. A holding company is generally inactive except from holding other companies. Whereas a parent company, usually has its own business ventures and purchases its subsidiaries for investment purposes.
If the parent company fails, it can pay its liquidation from its own bills as it is its own company. But if they own parts in the subsidiary, they can legally take liquidation from the subsidiary if they require it, which could affect the running of that business. This generally happens when the parent company can’t borrow enough money to pay of its debts.