If you mean

1. are the Julia sets that correspond to parameter values of the Mandelbrot set connected, then I do believe that both Julia and Fatou proved this (in 1918 and 1916 respectively).
2. are the Julia sets that correspond to parameters values from the interior of the Mandelbrot set locally connected? The definition of local connectivity is post 1918... This is indeed in the Orsay notes, but Douady and Hubbard were quite coy about who wrote what. Most of the results are joint (and some are by the people in the seminar).
3. are the Julia sets that correspond to parameters values from the interior of the Mandelbrot set locally connected? This is false, in the sense that some are, and some aren't. A whole slew of people provided useful results here, Siegel, Cramer, Douady-Hubbard, Yoccoz and more. It is a (pun intended) rather complex story.

If you mean

1. are the Julia sets that correspond to parameter values of the Mandelbrot set connected, then I do believe that both Julia and Fatou proved this (in 1918 and 1916 respectively).
2. are the Julia sets that correspond to parameters values from the interior of the Mandelbrot set locally connected? The definition of local connectivity is post 1918... This is indeed in the Orsay notes, but Douady and Hubbard were quite coy about who wrote what. Most of the results are joint (and some are by the people in the seminar).
3. are the Julia sets that correspond to parameters values from the boundary of the Mandelbrot set locally connected? This is false, in the sense that some are, and some aren't. A whole slew of people provided useful results here, Siegel, Cramer, Douady-Hubbard, Yoccoz and more. It is a (pun intended) rather complex story.