MSPs on the finance and public administration committee are seeking answers from the cross-party Scottish Parliamentary Corporate Body (SPCB), which manages the parliament and is chaired by Presiding Officer Alison Johnstone.
The website was revamped to comply with rules around readability and accessibility and is supposed to work better on mobile devices.
However, there have been complaints about its functionality.
She called it "an affront to the Scottish taxpayer", adding: "I was shocked when I received a response to my initial query about the entire cost of the replacement, given I have had constituents complain to me that it is a poorer version of the old website.”
SNP MSP Kenneth Gibson, convener of the finance and public administration committee, has now written to Ms Johnstone over the issue.
He said the committee wants to know "how the figure of £3,038,000 has been arrived at, including breakdowns for each financial year, and by phase, and details of capital and licensing costs and other relevant details".
Mr Gibson also asked if the project was delivered on budget, what tendering process was in place, what the ongoing costs of maintenance and improvement are, the reasons the revamp took so long and whether the SPCB thought it "value for money", among other questions.
He added: "We look forward to receiving a comprehensive response to the issues raised in this letter by April 26.”