This is a classical type of task where you use recursion. Recursion can be tricky as it is, but especially so when dealing with tinyxml - not the friendliest API.
STEP 1: A find Helper
Let's make it friendlier. At all levels we will want to visit all "entity" elements. Let's make a handy helper to use TinyXML to get those:
auto find(TiXmlElement const* node, char const* name) {
    std::vector<TiXmlElement const*> found;
    for (
            auto el = node->FirstChildElement(name);
            el;
            el = el->NextSiblingElement(name)
        )
    {
        found.push_back(el);
    }
    return found;
}
So far so good, now we can more easily query nodes with a certain name.
STEP 2: Parse a Single Entity
Let's forget about that helper for a second, and assume we already have an "entity" node selected. Now we want to parse it into an Entity (name, sprite, etc.) and return a newly created entity:
Entity* parse_entity(TiXmlElement const* node) {
    Entity* entity = g_manager.createEntity(node->Attribute("name"));
    // todo transforms, sprite info etc.
    return entity;
}
STEP 3: Build The Tree
That sounds like the complicated part. But really, using our find helper from above, all we need is this:
void parse_sub_entities(TiXmlElement const* node, Entity* parent = nullptr) {
    for (auto el : find(node, "entity")) {
        auto entity = parse_entity(el);
        if (parent && entity) {
            entity->addParent(parent);
            parent->addChild(entity);
        }
        parse_sub_entities(el, entity);
    }
}
All child nodes are parsed using parse_entity from Step #2, and only if we had a parent node we add the relationships.
To finish it all up, we recurse into child entities, this time passing the current entity as the parent.
Full Demo
**Not Live On Coliru**¹
#include <tinyxml.h>
#include <vector>
#include <list>
#include <iostream>
#include <iomanip>
namespace { // helper functions for XML searching
    auto find(TiXmlElement const* node, char const* name) {
        std::vector<TiXmlElement const*> found;
        for (
                auto el = node->FirstChildElement(name);
                el;
                el = el->NextSiblingElement(name)
            )
        {
            found.push_back(el);
        }
        return found;
    }
}
auto scene = R"(<scene>
    <entity name="Parent Entity">
        <transform posx="500" posy="100" scalex="1" scaley="1" rotation="0"/>
        <sprite image="Assets/Sprites/Avatar2.png"/>
        <entity name="Child Entity">
            <transform posx="0" posy="0" scalex="1" scaley="1" rotation="0"/>
            <sprite image="crimson-logo.png"/>
        </entity>
    </entity>
</scene>)";
struct Entity {
    std::string _name;
    Entity* _parent = nullptr;
    std::vector<Entity*> _children;
    explicit Entity(std::string name = "unnamed") : _name(std::move(name)) {}
    void addChild(Entity* e) { _children.push_back(e); }
    void addParent(Entity* e) {
        assert(!_parent || _parent == e);
        _parent = e;
    }
};
struct Mgr {
    std::list<Entity> _entities;
    Entity* createEntity(std::string name) {
        return &_entities.emplace_back(name);
    };
} g_manager;
Entity* parse_entity(TiXmlElement const* node) {
    Entity* entity = g_manager.createEntity(node->Attribute("name"));
    // todo transforms, sprite info etc.
    return entity;
}
void parse_sub_entities(TiXmlElement const* node, Entity* parent = nullptr) {
    for (auto el : find(node, "entity")) {
        auto entity = parse_entity(el);
        if (parent && entity) {
            entity->addParent(parent);
            parent->addChild(entity);
        }
        parse_sub_entities(el, entity);
    }
}
int main() {
    TiXmlDocument doc;
    doc.Parse(scene);
    parse_sub_entities(doc.RootElement());
    std::cout << "Manager has " << g_manager._entities.size() << " entities\n";
    for (auto& e: g_manager._entities) {
        std::cout << "==== Entity: " << std::quoted(e._name) << "\n";
        if (e._parent)
            std::cout << " - has parent " << std::quoted(e._parent->_name) << "\n";
        for (auto child : e._children)
            std::cout << " - has child " << std::quoted(child->_name) << "\n";
    }
}
Prints:
Manager has 2 entities
==== Entity: "Parent Entity"
 - has child "Child Entity"
==== Entity: "Child Entity"
 - has parent "Parent Entity"
¹ no tinyxml installed on any online compiler I know