Hello @phoenix52,
Yes you can transfer unlisted shares to your parents as gift and capital gain earned on them shall be taxed in the hands of transferee. (Please check your company’s policy if transfer is permissible or restricted)
Your understanding for LTCG/ STCG taxation is correct, but HUF shall not be considered as your “relative” as per income tax act. So if value shares transferred to your father’s HUF is more than INR 50K then tax under IFOS shall be levied on FMV of shares on date of transfer in hands of HUF.
Hope this helps!