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The requirement is to calculate and display a roll-up of child and "grandchild" records (cases of the account and its parent) where a standard master-detail relationship does not exist. This requires custom automation. The combination of Process Builder and Flow is a powerful declarative solution for this.

A process on the Case object can be triggered when a case is created or its status is updated. This process can then invoke an autolaunched Flow. The Flow is better suited for the complex logic of querying all open cases for both the direct account and the parent account, performing the count, and then updating custom fields on the original Account record.

B. Workflow: Workflow Rules are a legacy tool that cannot query and aggregate data from multiple unrelated records, which is necessary to count cases.

C. Approval Process: Approval Processes are used exclusively for formal record review and approval sequences and cannot perform data aggregation or roll-up calculations.

1. Salesforce Help, Invoke a Flow from a Process: This official document describes the design pattern for this solution. It states, "The combination of processes and flows is powerful. Processes can start when a record is changed and can perform some actions. But for more complex logic, a process can pass the work to a flow." This directly supports using Process Builder as a trigger for a Flow that handles the aggregation logic.

2. Salesforce Help, Flow Element: Get Records: This document details the capability of Flow to find and retrieve multiple Salesforce records that meet specific criteria. This element is essential for gathering all the open Case records related to both the account and its parent account to perform the count.

3. Salesforce Architects, Record-Triggered Automation Decision Guide: This guide advises on choosing the correct automation tool. For requirements that involve querying for and acting upon other records (like counting related cases), it recommends Flow as the appropriate tool for the logic, which can be triggered by a process.