Looks like you're passing in partner as a prop, so if you have the tasks for that partner on the partner object through an Eloquent relationship you could just use that in your Vue data:
export default {
props:['partner'],
data(){
return{
task: null,
tasks: this.partner.tasks,
}
},
methods:{
save(){
if(this.task){
this.pushToTasks(this.task);
axios.post(`/tasks`, {
task: this.task,
})
this.task = null;
}
},
pushToTasks(task) {
this.tasks.push({
task: task,
});
},
}