I had luck some time ago using TM GUI. TM seems to run with different privileges than user... Hold option and click on TM icon in the bar - and "Browse other backup disks..." appears. Navigate to old TM backup disk.
If that fails, you :
1. May be able to access files when running as admin user. Log in the system as admin and try.
2. Might be able to change ownership of specific folder you wanty to access... May be. Do get Info on the folder you want to access - cmd-I - on the folder with data and change access permissions. You will need admin privileges on the computer, but you can change who has read, write, and read/write privileges there. You need to unlock the little lock in lower right corner and then add user - or group - to have right access you want. You want to apply it to all inside content of that folder. If this is simply some folder you need access to, it should work. But this may break any chances to access that backup via TM interface!!!