Do you need your partner's permission to date someone new?
Not in well-functioning polyamorous structures. You need communication and respect for any agreed protocols, but new partners are not subject to existing-partner approval — that would be veto power, which is widely critiqued. Agreements about disclosure and pacing are the standard, not permission.
The framing 'permission' tends to import an assumption from a different structure. In polyamory, partners do not generally hold approval authority over each other's new connections — that would be veto power, which is widely critiqued in contemporary writing. What partners do hold is mutual agreement to specific protocols, which differ across polycules but typically include disclosure rather than approval.
Common protocols. Tell each other when you have started dating someone new (after, say, the first or second date, not before). Tell each other when you have started sleeping with someone new, especially if barriers have or have not been involved. Consult on shared-life-affecting decisions (introducing a new partner to children, having a new partner stay overnight in a shared home). None of these involve permission; they involve information-sharing and joint decision-making about shared infrastructure.
Where the line is between communication and permission. Information that affects the existing partner's life or wellbeing should be shared so they have agency over their own response — they can adjust their own behaviour, raise concerns, make decisions about their own relationship — based on real information. That is communication. Information that gates whether you can have the relationship at all is permission, and it is structurally similar to veto.
What healthy practice looks like. You have ongoing connections you have not closed off. When something happens (a meaningful new connection, a sexual encounter that crosses some agreed line, anything that would affect the existing partner), you tell them. They have the right to react, to ask questions, to adjust how they show up. They do not have the right to require you to undo the new connection — and you do not have that right over them.
Where this gets harder. New polyamorous structures often have permission-like agreements baked in (mutual veto, for instance). These tend to get re-examined as the structure matures and the practitioners realise the agreement is producing patterns they don't actually want. The slow move from permission-based structures to communication-based structures is one of the predictable arcs of polyamorous practice.