I can't find any information https://www.gov.uk/start-up-visa/your-partner-and-children is mum on this.
Summary (Jan 2022): Yes, the dependent partner of a UK Start-up visa holder is allowed to work, subject to some exceptions (which applies to majority of work visas).
According to the Immigration Rules (last updated 4 Jan 2022), in particular Immigration Rules Appendix Start-up, the dependent partner of a UK Start-up visa holder is allowed to work if they are themselves granted a visa (emphasis and correction on the text formatting around SU 22.3(a) my own):
Period and conditions of grant for a dependent partner and dependent child on the Start-up route
SU 22.1. A partner will be granted permission which ends on the same date as their partner’s permission on the Start-up route
SU 22.2. [...]
SU 22.3. The grant will be subject to all the following conditions:
(a) no access to public funds; and
(b) work (including self-employment and voluntary work) permitted; except for employment as a professional sportsperson, including as a sports coach; and
(c) study is permitted, subject to the ATAs [sic] condition in Appendix ATAS; and
(d) if Part 10 applies the applicant will be required to register with the police.