Add Pterodactyl docs (#38158)

* Add docs for Pterodactyl

* Fix findings of coderabbitai

* Fix findings of coderabbitai (part 2)

* Correct ha_release

* Replace Host with URL

* Add brief description about the integration itself and not only about Pterodactyl in general

* Rename client API key to account API key and fix review findings

* Fix remaining review finding

source/_integrations/pterodactyl.markdown

@@ -0,0 +1,58 @@
+---
+title: Pterodactyl
+description: Instructions on how to integrate a Pterodactyl server into Home Assistant.
+ha_release: 2025.4
+ha_category:
+  - Binary sensor
+ha_iot_class: Local Polling
+ha_config_flow: true
+ha_codeowners:
+  - '@elmurato'
+ha_domain: pterodactyl
+ha_platforms:
+  - binary_sensor
+ha_integration_type: integration
+ha_quality_scale: bronze
+---
+
+[Pterodactyl](https://www.pterodactyl.io) is a game server management panel designed to simplify the administration of game servers. It offers a user-friendly interface  which allows users to manage multiple game servers from a single dashboard, supporting popular games like Minecraft. Its key features include an intuitive web-based control panel, automated server installation, real-time server monitoring, scheduled backups and more. Each game server runs in an isolated Docker container, ensuring security and stability.
+
+The Pterodactyl {% term integration %} allows you to monitor your game servers of your Pterodactyl server within Home Assistant.
+
+## Prerequisites
+
+To access your Pterodactyl server, an account API key is required. Follow these steps to create a new one:
+
+- Access your **Pterodactyl Panel** and log in with your user account.
+- From the main dashboard, click your **username** or **profile icon** in the top-right corner, then select **API Credentials**.
+- Enter a **Description** to identify the key (for example, "Home Assistant").
+- Optionally, specify **Allowed IPs** to restrict where the key can be used (leave blank to allow all IPs).
+- Click **Create**. The panel will generate and display your new account API key.
+- Copy the **account API key** immediately and store it securely. You won’t be able to view it again after leaving the page.
+
+{% important %}
+Pterodactyl has two different types of API keys: Account and Application. Application API keys are not supported, an account API key as described above is required instead.
+{% endimportant %}
+
+{% include integrations/config_flow.md %}
+
+During setup, you will be prompted to enter the **URL** and the **account API key** of the server.
+
+{% configuration_basic %}
+URL:
+    description: "The URL of your Pterodactyl server, including the protocol (`http://` or `https://`) and optionally the port number. Example: `http://192.168.0.123:8080`"
+Account API key:
+    description: "The account API key for accessing your Pterodactyl server (see prerequisites)."
+{% endconfiguration_basic %}
+
+## Binary sensors
+
+This integration provides a binary sensor with the following information for each game server of your Pterodactyl server:
+
+- Status: `Running` or `Not running`
+
+## Removing the integration
+
+This integration follows standard integration removal. No extra steps are required.
+
+{% include integrations/remove_device_service.md %}