Lord is called as Parshuhast, as he is holding an axe weapon to cut the worldly tree. In spite of a worn condition, this axe is able to fully destroy the seed of transmigration.
Lord’s hands adorned with an axe, indicate the path of detachment. Attachment is the cause of worldly bondage. A pleasure-seeker is entangled in the cycle of creation due to his greed, but an ascetic crosses over this material ocean due to his detachment. Bhagwan Shankaracharya also repeatedly emphasizes the importance of detachment.
Gita also specifies the detachment as a weapon, by which one can root out the tree of birth and death—Asanga shastrena dridhen chittwa. (Chapter 15, text 3). Actions and prayers may be efficient to any extent, but they help us to reach to higher planets only, yet the transmigration is not stopped. This cycle of rebirth is stopped through the process of self-realization only. When we identify the detachment of Lord Shiva, then we also stay aloof from sense objects. We are non-different to Shiva, rather we are Shiva in our real nature.
Bhaskar Roy, an scholar of Vedic scriptures, explains the purport of this name as under—

अमोघचरितत्त्वं व्यनक्ति परशुहस्ते।
शिवागमवचोभिस्ततः परशुहस्तः॥

Purport: O Lord Shiva! Scriptures declare that your axe-weapon shows your infallible nature. You are called Parshuhasta as you are infallible. It means the mind, speech or physical actions of the Lord is never unrewarding, because he is Sat-Sankalpa (wish fulfilling). The infallible nature of Shiva is innate due to his disposition of pure goodness. In spiritual science, cutting the tree of dualism by the knowledge of Vedanta, is the true sense of worship of Parshuhasta.