Set up equations

eq1=x'[t]==σ(y[t]-x[t])
eq2=y'[t]==r x[t] - y[t] - x[t] z[t]
eq3=z'[t]==x [t]y[t]-b z[t]

Solve equations

sol1=With[{σval=10,bval=8/3,rval=28},NDSolve[{eq1,eq2,eq3,x[0]==0,y[0]==1,z[0]==0}/.{r->rval,σ->σval,b->bval},{x[t],y[t],z[t]},{t,0,1000}]];
sol2=With[{σval=10,bval=8/3},ParametricNDSolve[{eq1,eq2,eq3,x[0]==0,y[0]==1,z[0]==0}/.{r->rval,σ->σval,b->bval},{x,y,z},{t,0,50},rval]];

Plot with variable time

With[{endanimate=50},Manipulate[Row[{Show[Plot[{x[t]/.sol1,y[t]/.sol1,z[t]/.sol1},{t,0,tf},ImageSize->{400,Automatic},Ticks->None,PlotRange->{{0,endanimate},{-20,50}},PlotStyle->Thickness[0.002],PlotLabel->Style["The Lorenz equations σ=10,b=8/3,r=28",FontSize->20],PlotLegends->Placed[LineLegend[{"x","y","z"},LegendLayout->"Row"],{0.2,0.9}]],ListPlot[{{t,#/.sol1[[1]]}}&/@{x[t],y[t],z[t]}/.t->tf]],Show[ParametricPlot3D[{x[t],y[t],z[t]}/.sol1,{t,0,tf},ImageSize->{400,Automatic},Ticks->None,PlotStyle->Directive[Black,Thin],PlotPoints->100,PlotRange->{{-20,30},{-30,50},{-10,50}},AxesLabel->(Style[#,FontSize->20]&/@{"x","y","z"})],ListPointPlot3D[{{x[t],y[t],z[t]}/.sol1[[1]]/.t->tf},PlotStyle->Black]]}],{{tf,1,Style["Simulation time",FontSize->20]},1,endanimate}]
]

Plot with variable r

With[{tf=50},Manipulate[Row[{Show[Plot[{x[rval][t]/.sol2,y[rval][t]/.sol2,z[rval][t]/.sol2},{t,0,tf},ImageSize->{400,Automatic},Ticks->Automatic,PlotRange->{{0,tf},Automatic},PlotStyle->Thickness[0.002],PlotLegends->Placed[LineLegend[{"x","y","z"},LegendLayout->"Row"],{0.2,0.9}]],ListPlot[{{t,#/.sol1[[1]]}}&/@{x[t],y[t],z[t]}/.t->tf]],Show[ParametricPlot3D[{x[rval][t],y[rval][t],z[rval][t]}/.sol2,{t,0,tf},ImageSize->{400,Automatic},Ticks->None,PlotStyle->Directive[Black,Thin],PlotPoints->100,PlotRange->{{-20,30},{-30,50},{-10,50}},AxesLabel->(Style[#,FontSize->20,FontFamily->"EB Garamond"]&/@{"x","y","z"})],ListPointPlot3D[{{x[rval][t],y[rval][t],z[rval][t]}/.sol2[[1]]/.t->tf},PlotStyle->Black]]}],{{rval,0.1,Style["r",FontFamily->"EB Garamond",FontSize->20]},0.1,30}]
]