There is no nothing wrong in
supplicating to Allah for yourself and your parents after invoking Allah’s peace and blessings upon the Prophet Muhammad (peace be upon him) in the last Tashahhud. However, this should be done after asking Allah’s Refuge against the torture of Hell-fire, (Part No. 5; Page No. 402) the punishment in the grave, the trials of life and death, and the trial of Al-Masih-ul-Dajjal (the Antichrist). The Prophet (peace be upon him) urged the believers to exert themselves in supplicating to Allah during Tashahhud and did not specify a Du‘a’ (supplication). Moreover, Allah (Exalted be He) says:
...and ask forgiveness for your sin, and also for (the sin of) believing men and believing women.
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.