I think the problem is the horizontal scales of the two plots, which are completely different. I would suggest working with absolute time in both the fit and the plot of the fitting parabola:

Clear[datedWeights, fit, weights, fitPlot];
datedWeights = {{"2016-06-02", 84.4}, {"2016-06-03", 
    83.7}, {"2016-06-04", 82.7}};
weights = {AbsoluteTime[#[[1]]], #[[2]]} & /@ datedWeights;
fit[x_] = Fit[weights, {1, x, x^2}, x];
fitPlot = 
  DateListPlot[
   Table[{x, fit[x]}, {x, AbsoluteTime[datedWeights[[1, 1]]], 
     AbsoluteTime[datedWeights[[-1, 1]]], 3600}], PlotStyle -> Red];
dateListPlot = DateListPlot[datedWeights, PlotRange -> All];
Show[{fitPlot, dateListPlot}]
Show[{dateListPlot, fitPlot}]