Try with LineLegend combined with LegendLayout -> "Row":

    Plot[{d[10, 1, 1, 2, t, 0], , , ,}, {t, 0, 20}, PlotRange -> {0, 5}, 
     LabelStyle -> {FontSize -> 22, FontFamily -> "Times", Black, Bold}, 
     PlotStyle -> {{Darker[Blue], 
        Thickness[0.004]}, {Darker[Blue]}, {Darker[Blue]}, {Darker[
         Blue]}, {Darker[Blue]}}, PlotRangePadding -> 0, 
     LabelStyle -> {FontSize -> 13}, AxesStyle -> Bold, Frame -> True, 
     FrameLabel -> {t[sec], \[ScriptCapitalD][t]}, 
     LabelStyle -> {FontSize -> 13}, AxesStyle -> Bold, 
     FrameStyle -> {{Directive[FontColor -> Darker[Red]], 
        Automatic}, {Directive[FontColor -> Darker[Red]], Automatic}}, 
     PlotLabel -> \[ScriptCapitalD] == 
       Subscript[\[ScriptCapitalD], 
         0] \[Theta][
          t - Subscript[t, 0]]/(1 + 
           Log["1+\[Alpha] t"] (c + Sin[\[Beta] t])), 
     PlotLegends -> 
      Placed[LineLegend[{Subscript[\[ScriptCapitalD], 0] == 0, 
         Subscript[t, 0] == 0, "\[Alpha]=1", "c=2", "\[Beta]=1"}, 
        LegendLayout -> "Row"], {0.3, 0.3}]
     ]