  We constrain the planetary atmospheric mass fraction at the time of the
dispersal of the protoplanetary disk and the evolution of the stellar rotation
rate for a dozen multi-planet systems that host sub-Neptunes and/or
super-Earths. We employ a custom-developed Python code that we have dubbed
Pasta (Planetary Atmospheres and Stellar roTation rAtes), which runs within a
Bayesian framework to model the atmospheric evolution of exoplanets. The code
combines MESA stellar evolutionary tracks, a model describing planetary
structures, a model relating stellar rotation and activity level, and a model
predicting planetary atmospheric mass-loss rates based on the results of
hydrodynamic simulations. Through a MCMC scheme, we retrieved the posterior
PDFs of all considered parameters. For ages older than about 2 Gyr, we find a
median spin-down (i.e. $P(t)\propto t^y$) of $\bar{y}=0.38_{-0.27}^{+0.38}$,
indicating a rotation decay slightly slower than classical literature values
($\approx$0.5), though still within $1\sigma$. At younger ages, we find a
median spin-down (i.e. $P(t)\propto t^x$) of $\bar{x}=0.26_{-0.19}^{+0.42}$,
which is below what is observed in young open clusters, though within
$1\sigma$. However, these two results are likely due to a selection bias as the
systems suitable to be analysed by Pasta contain at least one planet with a
hydrogen-dominated atmosphere, implying that the host star has more likely
evolved as a slow rotator. We further look for correlations between the initial
atmospheric mass fraction of the considered planets and system parameters, but
without finding any. The TESS, CHEOPS, and PLATO missions are going to be
instrumental in identifying and precisely measuring systems amenable to Pasta's
analysis and can thus potentially constrain planet formation and stellar
evolution.
