Loading from the user:// dir works as intended. I put spineboy-pro.skel and spineboy.atlas/spineboy.png into

/Users/badlogic/Library/Application Support/Godot/app_userdata/spine-godot-examples/save/animation

To simulate your setup. spine-godot-examples is the project name. In your case it's DesktopPets.

I then load the skeleton and atlas via this code:

extends Node2D

func _ready():
	# Load the skeleton file
	var skeleton_file_res = SpineSkeletonFileResource.new();
	var error = skeleton_file_res.load_from_file("user://save/animation/spineboy-pro.skel");
	if (error != OK):
		print("Could not load skeleton")	
	
	# Load the atlas file
	var atlas_res = SpineAtlasResource.new();
	error = atlas_res.load_from_atlas_file("user://save/animation/spineboy.atlas");
	if (error != OK):
		print("Could not load atlas")	
	
	# Create a skeleton data resource, you can share this across multiple sprites
	var skeleton_data_res = SpineSkeletonDataResource.new();
	skeleton_data_res.skeleton_file_res = skeleton_file_res;
	skeleton_data_res.atlas_res = atlas_res
	
	# Create a sprite from the skeleton data and add it as a child
	var sprite = SpineSprite.new();
	sprite.skeleton_data_res = skeleton_data_res;
	sprite.position.x = 200;
	sprite.position.y = 200;
	sprite.get_animation_state().set_animation("walk", true, 0);
	self.add_child(sprite)
	pass

This works as intended.