No, the quantization of negative charge as -1/3 does not imply that the electron has more "parts" or an inner structure. The charge of an electron is considered to be a fundamental, indivisible unit of -1 elementary charge.
Quarks, on the other hand, are elementary particles that possess fractional charges. Quarks have been observed to have charges of either +2/3 or -1/3 elementary charge. However, it's important to note that quarks are always found in combinations to form composite particles such as protons and neutrons, which have integral charges. Quarks are bound together by the strong nuclear force and cannot be observed in isolation.
Electrons, along with other leptons (such as muons and tau particles), are classified as point-like elementary particles with no known substructure. They are not composed of quarks or any other constituents. As far as our current understanding goes, electrons are considered to be fundamental particles with a charge of -1 elementary charge and no internal structure.