  We consider a $2\times 2$ system of parabolic equations with first and zeroth
coupling and establish a Carleman estimate by extra data of only one component
without data of initial values. Then we apply the Carleman estimate to inverse
problems of determining some or all of the coefficients by observations in an
arbitrary subdomain over a time interval of only one component and data of two
components at a fixed positive time $\theta$ over the whole spatial domain. The
main results are Lipschitz stability estimates for the inverse problems. For
the Lipschitz stability, we have to assume some non-degeneracy condition at
$\theta$ for the two components and for it, we can approximately control the
two components of the $2 \times 2$ system by inputs to only one component. Such
approximate controllability is proved also by our new Carleman estimate.
Finally we establish a Carleman estimate for a $3\times 3$ system for parabolic
equations with coupling of zeroth-order terms by one component to show the
corresponding approximate controllability with a control to one component.
