  We introduce the concept of finite $\gamma$-scaled quadratic variation along
a sequence of partitions for paths on a given interval. This concept, with
historical roots in the study of Gaussian processes by Gladyshev (1961) and
Klein \& Gin\'e (1975), includes the fractional Brownian motion (fBM) with
Hurst index $H$, which has finite $1-2H$-scaled quadratic variation. We show
that a path that is controlled by a path with finite $\gamma$-scaled quadratic
variation in the sense of M. Gubinelli inherits this property, and the
corresponding scaled quadratic variation satisfies an It\^o-isometry type
formula. Moreover, we prove quantitative error bounds that establish a
relationship between the convergence rates of the scaled quadratic variation of
the controlled path and that of the controlling path. Additionally, we
introduce a consistent estimator for the parameter $\gamma$ based on a single
sample path, complete with quantitative error bounds. We apply these results to
the parameter estimation for fractional diffusions. Our findings specify
convergence rates for the estimation of both the Hurst index and the parameters
in the noise vector fields. The paper concludes with numerical experiments that
substantiate our theoretical findings.
