How to get information from the both sides of @ManytoMany relationship?

Question

I'm new to JPA and I'm trying to build a @ManyToMany relationship between two entities: car.java and part.java. I wrote a method, which saves the relationship in a third table: car_part.

car entity:

 @ManyToMany(fetch = FetchType.EAGER)
    @JoinTable(name = "car_part", joinColumns = @JoinColumn(name = "car_id"),
    inverseJoinColumns = @JoinColumn(name = "part_id"))
    private Set<Part> parts = new HashSet<>();

part entity:

@JsonIgnore
@ManyToMany(mappedBy = "parts")
private Set<Car> cars = new HashSet<>();

car controller:

@PutMapping("/cars/{carId}/parts/{partId}")
    public Car addPartToCar(@PathVariable Long carId, @PathVariable Long partId) {
        
        Car car = carManager.findById(carId).get();
        Part part = partManager.findById(partId).get();
        
        car.addPart(part);

        return carManager.save(car);
    }

When I run a "getCar" method, a part is included:

{
    "id": 3,
    "brand": "vw",
    "numberPlate": "12345",
    "parts": [
        {
            "id": 2,
            "partName": "part1",
            "price": "200"
        }
    ]

But when I run a "getPart" method, a car is not included:

{
        "id": 2,
        "partName": "part1",
        "price": "200"
    }

My Question: Why is a car not included, when I'm searching for a part? I searched and I found out, I should add the relationship to the both sides. So I should probably add a car to the part like this, but I'm getting a LazyInitializationException

@PutMapping("/cars/{carId}/parts/{partId}")
    public Car addPartToCar(@PathVariable Long carId, @PathVariable Long partId) {

        Car car = carManager.findById(carId).get();
        Part part = partManager.findById(partId).get();

        car.addPart(part);
        part.getCars().add(car);

        partManager.save(part);

        return carManager.save(car);

    }

Any help you can give would be greatly appreciated.

UPDATE: I removed @JsonIgnore but now I get the HttpMessageNotWritableException