From a quick look at your problem, it seems that it can be solved in $O(nk)$ time using dynamic programming.
Let $[a_1, \dots, a_n]$ be your input array and define $\delta(i,j)$ as the minimum sum of the differences between the first and last element of each segment in an optimal subdivision of $[a_1, \dots, a_i]$ into at most $j$ segments. If no feasible subdivision exists then let $\delta(i,j) = +\infty$.
Then:
- $\delta(0,0)=0$;
- $\delta(i,0)=+\infty$ for $i>0$; and
- $\delta(i,j) = \min_{\substack{h = 1, \dots, i \\ a_i - a_h < 125}} \big\{ \delta(h-1, j-1) + a_i - a_h \big\}$, for $i,j>0$ (where the minimum over an empty set is $+\infty$).
The value of the optimal solution is $\delta(n, k)$ and, since computing each $\delta(i,j)$ requires $O(1)$ time, the overall time required is $O(nk)$. The optimal solution can be reconstructed using standard techniques (e.g., by retracing the optimal choices backwards, or by storing -for each $\delta(i,j)$- the optimal value of $h$ in the minimum).