可以使用reduce()和filter()通过children id获取子对象

const images = {
"data": [
    {
        "id": "1",
        "name": "Amazon",
        "image": "data:image/png;base64,iVBORw0KGgoAAA",
        "children": [
            {
                "id": "11",
                "name": "Clothes",
                "image": "data:image/png;base64,iVBORw0KG",
                "url": "http://www.google.com"
            },
            {
                "id": "12",
                "name": "Crockery",
                "image": "data:image/jpeg;base64,/9j/4AAQSkZJRgABAQAAAQ",
                "url": "http://www.google.com"
            },
            {
                "id": "13",
                "name": "Home Appliances",
                "image": "data:image/jpeg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wCEA",
                "url": "http://www.google.com"
            }
        ]
    }
]
};
const findImage = ((parent_id, child_id) => {
return result = images.data.reduce((r, o) => {
    const children = o.children.filter(
        ({ id }) => id === child_id && o.id === parent_id);
    if (children.length) {
        r.push(Object.assign({}, o, { children }));
    }
    return r;
}, [])[0];

})

console.log(JSON.stringify(findImage('1', '11'), null, 4));