Question

To clarify, can any parish councillor ask to view the council computer?

Answer 1

What would you want/need to see. Your question suggests you’ve asked for something and it’s not been shown or you have a suspicion.

For transparency reasons much of the info needed should be readily available on the council website for example.

GDPR will restrict you from viewing certain materials but in local government these should be quite few.
Can you be more specific about what you’re hoping to see?

Comments

GDPR 2018 does not restrict Cllrs from viewing "certain materials".  It places obligations on their Council collectively to manage any personal data they hold in accordance with seven principles, these being:

    Lawfulness, fairness and transparency
    Purpose limitation
    Data minimisation
    Accuracy
    Storage limitation
    Integrity and confidentiality (security)
    Accountability

Answer 2

The computer is (or should be) the property of the Council not the clerk.

David Merson posts a number of articles on the Local Government website https://www.localgovernmentlawyer.co.uk/governance/314-governance-a-risk-articles/15641-councillors-rights-of-access-to-information

He asserts that "In principle councillors have, in addition to the ordinary rights of access to certain information enjoyed by the general public, the right to access any information held by the council of which they are a member.  He also says "Under common law principles councillors have the right to access information held by their authority where it is reasonably necessary to enable the member to properly perform their duties as a councillor" He does add though that there is no single source of right on which to base a claim to entitlement to information or documents.  Equally, you would have to conclude there does not appear to be any definitive piece of legislation that prevents Cllrs accessing non exempt or non confidential information. 

Taking all  that into account I believe they can see information on the Council Computer unless the Clerk can cite specific legislation to the contrary.