USAGE OF परामृशन्
Here the root is मृश आमर्शने । आमर्शनं means स्पर्शः = to touch. When it is prefixed with परा, it may mean “to touch”, “to think”, “to reflect”, “to consider”, “to rub” etc. If it is prefixed with परा and suffixed with शतृ-प्रत्यय, it takes the form परा+मृश्+शतृ=परामृशन् ।
See the usage in Kumarasambhavam by Kalidasa - तेजोदग्धानि गात्राणि पाणिनास्य परामृशन्। Here, पाणिना परामृशन् means touching by hand. Also see the usage in Mahabharata तूणांश्चापि परामृशन् meaning touching the quiver. Observe this sentence, इष्टदेवतां परामृशन् meaning thinking the desired God. See this sentence - विवेकी दूरदर्शी यो नीरं क्षीरं परामृशन्। It means that thinking the difference between the water and the milk. Observe this sentence from Nitishastra by Masuraksha - कौशलं सुप्रयुञ्जीत वस्तुजातं परामृशन्। Here, परामृशन् means considering. Consider this sentence from Ramayana - शप्तो ह्येष पुरा पापो वधूं रम्भां परामृशन्. Here, परामृशन् means assaulting. Thus, परामृशन् means touching, thinking, considering, assaulting etc.
Now, we will take only the meaning of “touching” here. हस्तेन परामृशन् means touching by hand and पादेन परामृशन् means touching by leg. However, if you change the तृतीयाविभक्ति into द्वितीयाविभक्ति, the meaning entirely differs. See these sentences - हस्तं परामृशन्, पादं परामृशन्. Here, these sentences mean “touching the hand (not by hand)” and “touching the foot (not by foot)”. If you use तृतीयाविभक्ति, हस्तेन परामृशन् it means “touching by hand (some other thing)”. In the same way if use तृतीयाविभक्ति, पादेन परामृशन्, it means “touching by foot (some other thing)”.
In the same way if you use द्वितीयाविभक्ति, हस्तं परामृशन्, पादं परामृशन्, it means that “touching the hand” and “touching the foot”. I hope I made it clear.