A bit of a mixture of legislation involved here but the DPA 2018 s7(3) states that parish councils are not considered to be a public authority for the proposes of GDPR so do not need to appoint a DPO. However, the ICO advice is that all parish councils need to consider their own circumstances as to whether that should apply.
s112 in effect gives a parish councils the authority to appoint staff to carry out their functions so I would suggest that if, having considered the circumstances, your council feels a DPO is relevant to your functions, you have the power to appoint by virtue of s112.