Maximal closed connected subgroups of positive dimension in $\mathrm{SL}(n,\mathbb{R})$ are parabolic, the normalizer of a connected semisimple subgroup, or the normalizer of a maximal torus. There are finitely many conjugation classes of each type, and so you could try to work out the maximal ones.
General References
According to the MathSciNet review (see also MO answers here and here) this paper addresses the problem (without proof):
Komrakov, B. P. Maximal subalgebras of real Lie algebras and a problem of Sophus Lie. Dokl. Akad. Nauk SSSR 311 (1990), no. 3, 528--532; translation in Soviet Math. Dokl. 41 (1990), no. 2, 269–273 (1991)
Another paper that might be helpful is:
Mostow, G. D. On maximal subgroups of real Lie groups. Ann. of Math. (2) 74 1961 503–517.