Who authorizes a Royal Marines deployment

Can a British prime minister deploy the Royal Marines without separate parliamentary or royal approval?

Yes, in the explanation given: ministers can direct deployments without a separate approval vote for each one, while parliamentary convention and the Crown form part of the wider constitutional context.

Deployment authority is presented as a general power of the British government, applying to the Royal Marines and the other armed services. Parliamentary assent for military action is treated as a convention, separate from routine deployment authority. The armed forces' relationship with the Crown is a further constitutional consideration. A guest then imagines a monarch opposing a deployment, and the reply stresses the resulting constitutional complications. That hypothetical is a discussion of a possible crisis, rather than an established procedure for a royal veto.