How can I export all group policies to a easily-parseable format?

Unfortunately, with the modular nature of group policy extension handlers I don't believe you're going to find any tool better than Group Policy Management Console to do what you're after.

Your easiest path is going to be to either use a computer joined to the domain to run GPMC and examine / export the GPOs. Of course, if there are GPOs that have had their default permissions modified you might not have access to them w/ your credentials to audit them.

Given how easy it would be do to what you want with GPMC on a domain-member computer I'd say that you should pursue that. If you have problems, I'd approach it as a "political" / management problem and not a technical problem. GPMC is the right tool for the job, and if you're expected to do the job you need to have access to do it. The same would go with accessing GPOs that your domain credentials might not have access to.

If you absolutely can't get access to a domain-member computer running GPMC your next best (but undesirable alternative) would be to have an administrator there backup all the GPOs and give the backup to you. You could import that backup into another AD domain that you do control and audit the GPOs there. The problem with that strategy is that all SID information in the original domain will be incomprehensible to you in your own domain (and will be lost if you "map" the imported GPOs to users / groups inside your domain).

GPMC is your tool. Figure out how to make the "powers that be" allow you to use the tool and you'll get the job done quickly and efficiently.

Starting with Windows Server 2008 R2 or Windows 7 with RSAT installed you can use Powershell to export all your GPO settings either to HTML or to XML with Get-GPOReport.

Import-Module GroupPolicy

# Export a specific GPO
Get-GPOReport -Name MyFooGPO -ReportType Html -Path MyFooGPOReport.htm
Get-GPOReport -Name MyFooGPO -ReportType Xml -Path MyFooGPOReport.xml

# Export all GPOs
Get-GPOReport -All -ReportType Html -Path AllGPOsReport.htm
Get-GPOReport -All -ReportType Xml -Path AllGPOsReport.xml

There are many other helpful Cmdlets contained in the GroupPolicy module.