s = ParametricNDSolve[{x'[t] == y[t], y'[t] == x[t], x[0] == param,
    y[0] == param}, {x, y}, {t, 0, 1}, {param},
   WorkingPrecision -> 70, PrecisionGoal -> 20];
Plot[Evaluate[Table[x[param][t] /. s, {param, 1, 10}]], {t, 0, 1},
 PlotLegends -> Automatic]

Plot[Evaluate[ Table[x[param][t] /. s, {param, 1, 10}]], {t, 0, 1},
   PlotLabels->Range[10]]