Can a parent's new partner be given any legal role in decision-making for the child?

Not directly, at least not as a matter of automatic entitlement - decision-making responsibility under the Divorce Act or the Children's Law Reform Act is generally reserved for the child's legal parents, and a parent's new partner doesn't acquire any legal decision-making authority simply by living with the parent or being in a relationship with them, no matter how involved they become in the child's daily life. A parent can, of course, informally rely on a new partner's help with everyday matters, but that's different from the new partner having actual legal authority recognized by a court.

If a new partner takes on a genuinely long-term, parent-like role over time, it is possible for them to seek a formal legal role as a non-parent through a court application, similar to how a stepparent or another non-parent can seek parenting time, contact, or in limited cases decision-making responsibility - but this requires its own application and evidence of that settled relationship, rather than happening automatically through remarriage or cohabitation. Parents who want a new partner formally involved in decisions, and the other parent who may have concerns about that, should get legal advice about what's actually achievable rather than assuming either automatic inclusion or automatic exclusion.