my humble opinion is "mít pod palcem" means to control, to dominate.
Example: Mám to tady pod palcem. / I control everything here. (Express of domination, of power to smash everything with the thumb only, like an ant.)
But the truth is nobody knows what the poet was trying to say.
I hope it helps.
In the verse he songs: ...že jsem troufalcem, že chci mít pod palcem, tvou dlaň.
(Youtube; 1982 Jiří Korn - Píšu pastelkou; at time: 1:12)
Translated word by word: ...I am presumptuous I want to have your palm under my thumb.
The verse means he wants (I want to have your palm under my thumb) her as a man, poetically expressed, and he knows he dares (I am presumptuous) to think about her this way.
The palm can be perceived as something soft, loving and full of empathy, in the figurative sense of course. So in this case, you can imagine the palm something similar and close to her soul, or thing more practically and imagine it as herself.
The thumb is here used as a domination element. Have something under a thumb.
The song is very love song, very nice, soft and platonic. She does not want him (přes tvůj nezájem / despite you do not care about me). The singer loves her and trying to deal with his love by writing down his feeling by many types of equipment on many types of surfaces (concrete, a poster, moulding and so on). All poetically, of course. The whole song is about love and writing/expressing his feeling.
Despite all of this explanation, as have been here said many times, nobody can see into artists head. C.