Ion-acoustic solitary waves (IASWs) in plasma consisting of ions, positrons and superthermal electrons in two distinct temperatures have been studied. The reductive perturbation method (RPM) has been employed to derive the Korteweg–de Vries and modified KdV equation. Numerical and analytical studies show that compressive and rarefactive solitons exist for the selected parametric range depending on the spectral indexes, $κ (κ_{h},κ_{c})$ and their respective densities ($\nu,\mu$). It is found that spectral indexes ($κ_{h},κ_{c}$) and their relative densities have significant impact on the basic properties, i.e., amplitude and width as well as on the nature of IASWs. Variations of amplitude and width for the compressive and rarefactive solitary waves have been analysed graphically with different plasma parameters like spectral indexes of cold and hot electrons ($k_{c}, k_{h}$), their respective densities, ionic temperature ratio, positron temperature ratio as well as with the temperature ratio of the two-electron species. The amplitude of the compressive (rarefactive) solitary waves increases (decreases) on increasing $k_{h}$. However, the amplitude of the compressive (rarefactive) solitary waves decreases (increases) on increasing $k_{c}$. The investigations of such solitary waves may be helpful for the critical understanding of space where superthermal electrons with two different temperatures exist along with positrons and ions (e.g. Saturn’s magnetosphere, pulsar magnetosphere).