Serialization/Deserialization of a self-referencing class hierarchy without a parameterless constructor

Question

I have a hierarchical class ConversationModel that needs to be serialized/deserialized to/from JSON.

Note that the ConversationNode class does NOT have a parameterless constructor. The only non-default constructor is also marked as internal. All class properties are get-only. This is deliberate to keep the class immutable, stateless, and idiot-proof.

using System;
using System.Collections.Generic;
using System.Diagnostics.CodeAnalysis;
using Newtonsoft.Json;

public class ConversationModel
{
    public ConversationNode Node { get; }

    public ConversationModel ()
        => this.Node = new ConversationNode(this, null);

    public static JsonSerializerSettings JsonSerializerSettings
        => new JsonSerializerSettings
        {
            Formatting = Formatting.Indented,
            ReferenceLoopHandling = ReferenceLoopHandling.Ignore,
            PreserveReferencesHandling = PreserveReferencesHandling.All,
            TypeNameAssemblyFormatHandling = TypeNameAssemblyFormatHandling.Full,
            Converters = new [] { new ConversationModelJsonConverter(), },
        };

    public string Serialize ()
        => JsonConvert.SerializeObject(this, this.GetType(), ConversationModel.JsonSerializerSettings);

    public static ConversationModel FromJson (string json)
        => JsonConvert.DeserializeObject<ConversationModel>(json, ConversationModel.JsonSerializerSettings);
}

public class ConversationNode
{
    public ConversationModel Model { get; }
    public ConversationNode Parent { get; }
    public ConversationNodeList Nodes { get; }

    internal ConversationNode (ConversationModel model, ConversationNode parent)
    {
        this.Model = model ?? throw (new ArgumentNullException(nameof(model)));
        this.Parent = parent ?? throw (new ArgumentNullException(nameof(parent)));
        this.Nodes = new ConversationNodeList (model, parent);
    }
}

public class ConversationNodeList:
    List<ConversationNode>
{
    public ConversationModel Model { get; }
    public ConversationNode Parent { get; }

    public ConversationNodeList (ConversationModel model, ConversationNode parent)
    {
        this.Model = model;
        this.Parent = parent;
    }
}

The JsonSerializer, of course, expects a public, default, parameterless constructor. How can I get it to deserialize the model while keeping parent objects referenced properly as they are currently settable only through the constructor?

public sealed class ConversationModelJsonConverter:
    JsonConverter<ConversationModel>
{
    public override void WriteJson (JsonWriter writer, [AllowNull] ConversationModel value, JsonSerializer serializer)
    {
        // ???
    }

    public override ConversationModel ReadJson (JsonReader reader, Type objectType, [AllowNull] ConversationModel existingValue, bool hasExistingValue, JsonSerializer serializer)
    {
        // ???
        return (null);
    }
}

The question is:

1. Is it possible to write a custom JsonConverter to achieve this? If so, how?
2. Is there an easier way than having to implement a custom JsonConverter?