The edited question (and the answer to the first version) stilll leave me somewhat confused, so I'd suggest starting with concrete low-rank examples in order to focus better on the issues involved. (Sometimes the notation gets in the way, as in the switch from the $q$ in the header to the prime $p$; everything here can be done uniformly for any power $q$ of a fixed prime $p$.)

There is some useful older literature, starting with the thesis work which Steinberg did with Brauer in Toronto, which treats characters of $\mathrm{GL}_3(\mathbb{F}_q)$ and $\mathrm{GL}_4(\mathbb{F}_q)$: see his 1951 paper here. Soon afterwards J.A. Green worked out combinatorially the characters of all finite general linear groups, as noted in the question, though he needs a lot of careful notation involving partitions and the like: see his 1955 paper here.

For instance, in the case of $G=\mathrm{GL}_3(\mathbb{F}_q)$, there are (up to conjugacy) just two parabolic subgroups $P_1, P_2$ strictly between the standard Borel subgroup $B$ of upper triangular matrices and the full group $G$; these correspond to the two simple roots. The orders of all these groups are easily computed, along with the index of each parabolic in $G$. e.g., $$|G|=q^3(q-1)^3(q+1)(q^2+q+1),\: |P_i|=|B|(q+1) = q^3(q-1)^3(q+1)$$

Steinberg works out explicitly the degrees of all irreducible characters, which makes it easy to see how induction works for each type of character of $P_i$ arising from a Levi subgroup of type $\mathrm{GL}_2(\mathbb{F}_q)$ times the multiplicative group of $\mathbb{F}_q$. The only extra complication here is the fact that such a Levi subgroup has a family of characters of degree 1 which are trivial on the derived (=special linear) group.

In any case, the study of such characters doesn't depend on introducing geometric interpretations of the parabolic subgroups. Concerning older literature, in his book on symmetric functions (either editin) Macdonald included a treatment of Green's work, which Springer reformulated in his lectures at IAS (published in Springer Lecture Notes 131). By 1976 the more sophisticated methods of Deligne-Lusztig had paved the way for determination of characters of all finite groups of Lie type, along lines Macdonald and Springer had conjectured. But general linear groups can be done using only combnatorial methods, due to some absence of "cuspidal" characters.

The edited question (and the answer to the first version) stilll leave me somewhat confused, so I'd suggest starting with concrete low-rank examples in order to focus better on the issues involved. (Sometimes the notation gets in the way, as in the switch from the $q$ in the header to the prime $p$; everything here can be done uniformly for any power $q$ of a fixed prime $p$.)

There is some useful older literature, starting with the thesis work which Steinberg did with Brauer in Toronto, which treats characters of $\mathrm{GL}_3(\mathbb{F}_q)$ and $\mathrm{GL}_4(\mathbb{F}_q)$: see his 1951 paper here. Soon afterwards J.A. Green worked out combinatorially the characters of all finite general linear groups, as noted in the question, though he needs a lot of careful notation involving partitions and the like: see his 1955 paper here.

For instance, in the case of $G=\mathrm{GL}_3(\mathbb{F}_q)$, there are (up to conjugacy) just two parabolic subgroups $P_1, P_2$ strictly between the standard Borel subgroup $B$ of upper triangular matrices and the full group $G$; these correspond to the two simple roots. The orders of all these groups are easily computed, along with the index of each parabolic in $G$. e.g., $$|G|=q^3(q-1)^3(q+1)(q^2+q+1),\: |P_i|=|B|(q+1) = q^3(q-1)^3(q+1)$$

Steinberg works out explicitly the degrees of all irreducible characters, which makes it easy to see how induction works for each type of character of $P_i$ arising from a Levi subgroup of type $\mathrm{GL}_2(\mathbb{F}_q)$ times the multiplicative group of $\mathbb{F}_q$. The only extra complication here is the fact that such a Levi subgroup has a family of characters of degree 1 which are trivial on the derived (=special linear) group.

In any case, the study of such characters doesn't depend on introducing geometric interpretations of the parabolic subgroups (though Steinberg does take advantage of some 2-transitive permutation characters in his paper). Concerning older literature, in his book on symmetric functions (either editin) Macdonald included a treatment of Green's work, which Springer reformulated in his lectures at IAS (published in Springer Lecture Notes 131). By 1976 the more sophisticated methods of Deligne-Lusztig had paved the way for determination of characters of all finite groups of Lie type, along lines Macdonald and Springer had conjectured. But general linear groups can be done using only combnatorial methods, due to some absence of "cuspidal" characters.