I have >$10k tuition fee carry forward, yet the maximum transfer to parent is only $2K. Why can't I increase the transfer amt to $5k (decreasing carry forward by $3k)?

In order to answer you, TurboTax has an FAQ link, included below, that answers multiple tuition related questions. 

Frequently asked questions about transferring and carrying forward tuition and education amounts

All parts of your question can be answered through this link by scrolling down to find these titles:

• How much can the student transfer?
• How much should the student transfer?
• Can the student transfer amounts that he or she carried forward from a previous year?