Sometimes. A foreign company with no office in India generally does not need GST registration merely for exporting goods or services to Indian businesses. But registration is required in specific situations: supplying online services (OIDAR) to unregistered Indian consumers, holding exhibitions or supplying goods from a place in India, or operating through an Indian establishment.
Conversely, if your Indian subsidiary provides services to its foreign parent, those can qualify as zero-rated exports — with proper agreements, invoicing, and LUT filing — which is one of the most common structures we set up for India–US groups.
The classification depends on the exact supply chain, so map it before invoices start flowing.