For the question in the title there can not be any answer because as far as I know, the Vedas don't talk about temples.
So, I am answering the following question:
As per Parashara smirti, Shudras were considered less pure to perform
some auspicious activities.
Is there any vedic statement which supports this claim?
Yes, we have some statement ...
The Veda talk about spiritual matters only, occasionally giving references to the other practices prevalent at that point of time.
The core values of Sanatana Dharma of Vedic era advocate self enquiry only. Temple, texts like Puranas, do not represent Sanatana Dharma.
The article on Indus Valley Civilisation (3300 BCE to 1300 BCE), which was considered ...