Based on the settlement agreement, each party agrees to pay equal share of private school tuition fees and extracurricular activity expenses. Does the parent with the sole legal custody has the full authority to decide on what school and activities are chosen without consulting the other parent?

Only if you are going to pay the full bill, 50% liability give each 50% rights on deciding.

Reason is some vindictive parents would send their kid to the most expensive school possible out of spite. Thus, this keeps things honest and in the kids best interest as well.

Based on the settlement agreement, each party agrees to pay equal share of private school tuition fees and extracurricular activity expenses. Does the parent with the sole legal custody has the full authority to decide on what school and activities are chosen without consulting the other parent?

Your court order should outline what rights you do - or don't - have. If you have sole legal custody, in most cases, that means you would have the right to enroll the children without consulting the other parent... but I'd double-check your paperwork just to be positive. My husband's divorce paperwork stated that they share joint legal custody and outlined which decisions were included with that.

Are you positive that only one parent has legal custody? That term is often misunderstood. For example: my state refers to "physical custody" as "placement" and "legal custody" as "custody." You may want to verify what your rights are before doing anything.