Bhagwan Parshvanath’s birthplace is the city of Varanasi.

In Jain scriptures and tradition, this city is mainly known by the ancient name Kāshi (Kashi), and also Banaras. So, for his Janma Kalyanak (birth), the place-name is given as:

• Kashi / Varanasi (Banaras) – capital of the Kashi kingdom
• Father: Raja Ashvasen
• Mother: Rani Vāmādevī (Vamadevi) ( jainknowledge.com)

So, the name of Varanasi at the time of Parshvanath Bhagwan’s birth is traditionally referred to as Kashi, which is the same city now called Varanasi/Banaras.

You can read a simple story related to Varanasi Parshvanath here