First I would recommend using the library "Microsoft XML, vx.x" where x stands for the newest version available to you (should be 6.0). After that you are good to go. But the things to consider when working with XML in VBA are to broad to cover in an answer. I will provide the code here to get your desired output though:
Sub parse_xml()
Dim XmlFileName As String: XmlFileName = "C:\Path\Filename.xml"
Dim XmlDocument As New MSXML2.DOMDocument60
Dim NodeA As IXMLDOMNode
Dim NodeB As IXMLDOMNode
Dim NodeC As IXMLDOMNode
XmlDocument.Load XmlFileName
For Each NodeA In XmlDocument.DocumentElement.ChildNodes
For Each NodeB In NodeA.ChildNodes
For Each NodeC In NodeB.ChildNodes
Debug.Print NodeC.ParentNode.ParentNode.BaseName & NodeC.ParentNode.BaseName & NodeC.BaseName
Next NodeC
Next NodeB
Next NodeA
End Sub
Please note that I had to expand your file a little bit in order to make it a valid XML-File which can be loaded into the XmlDocument variable. The Xml-File I used for this example is the following:
<?xml version="1.0" encoding="UTF-8"?>
<data>
<A1>
<B1>
<c1>value</c1>
<c2>value</c2>
<c3>value</c3>
</B1>
<B2>
<c1>value</c1>
<c2>value</c2>
<c3>value</c3>
</B2>
</A1>
<A2>
<B1>
<c1>value</c1>
<c2>value</c2>
<c3>value</c3>
</B1>
<B2>
<c1>value</c1>
<c2>value</c2>
<c3>value</c3>
</B2>
</A2>
</data>
This will generate the following output (I just used Debug.Print as you can see):
A1B1c2
A1B1c3
A1B2c1
A1B2c2
A1B2c3
A2B1c1
A2B1c2
A2B1c3
A2B2c1
A2B2c2
A2B2c3
Side note at the end: Different from what Pierre commented I think parsing XML in VBA is quite the achievable task. From my experience you only need the library I recommended at the top of the post to do the actual parsing of the file. I have worked with fairly complex XML-Files using only that library for parsing.
Edit: This is the code, that will only extract values for your desired A criterion:
Sub parse_xml()
Dim XmlFileName As String: XmlFileName = "C:\Path\Filename.xml"
Dim XmlDocument As New MSXML2.DOMDocument60
Dim NodeA As IXMLDOMNode
Dim NodeB As IXMLDOMNode
Dim NodeC As IXMLDOMNode
Dim DesiredA As String: DesiredA = "A2" 'Enter your desired A here
XmlDocument.Load XmlFileName
For Each NodeA In XmlDocument.DocumentElement.ChildNodes
If NodeA.BaseName = DesiredA Then 'This is the new line that selects your desired A
For Each NodeB In NodeA.ChildNodes
For Each NodeC In NodeB.ChildNodes
Debug.Print NodeC.ParentNode.ParentNode.BaseName & NodeC.ParentNode.BaseName & NodeC.BaseName
Next NodeC
Next NodeB
End If 'End of the If-Statement
Next NodeA
End Sub