No, the employee of a subsidiary company is not an employee of the parent company.
It is just semantics.
Only wondering, than.... do the pay cheques come from the parent or the subsidiary company?
One works for the outfit that authorises the cheques. Yes?
One lad who worked for KLM subsidiary for a time told me that when he reached the retirement age he has received full KLM ret. package. As if he worked directly for the company. Does BA work different?