The term parent company refers to a business entity that owns enough stock in another firm - so much so that the former has the ability to influence or control the management and operations of the latter, much like the relationship between parent and child. The firm that is under the control or influence of the parent company is called a subsidiary.
The relationship between parent companies and subsidiaries is not one to one. In fact, it is common practice for parent companies to have more than one subsidiary. If one takes the time to analyze companies all over the world, it would not be a surprise to find out that in certain industries, most of the companies are owned by a single parent company. At the end of the day, one will probably see that the industry is operated by only a few parent companies, with the individual smaller companies being mere subsidiaries.
Subsidiaries are formed in a variety of ways. One way by which a subsidiary is formed is by outright creation. This is done by a parent company establishing a new one. If the parent company is the sole owner of the subsidiary - without minority shareholders - it is called a wholly owned subsidiary. Another way by which a subsidiary is formed is via a takeover. If the parent company is able to control at least 51% of the shares of another company, the latter becomes a subsidiary. A takeover may be hostile or not, but the end result is the same: the company that was taken over becomes a subsidiary under the control of the parent company.