A holding company is a business that exists for the purpose of owning other companies and typically does not produce goods or offer services other than through its subsidiaries.
Holding companies are one type of parent company. However, a parent company often has its own purpose and may produce products and services under its own name. A parent company often acquires subsidiaries as investments or expansions into new areas of business. In other cases, the purchase is intended to help the with particular areas of the company's existing business.
Here's an example to explain parent companies vs. holding companies:
Google acquired Nest Labs, a home automation company, in January 2015. At that time, Google became a parent company to Nest, its subsidiary. Because Google had many other business ventures operating under its own name, it was not a holding company.