How do I design a DP algorithm to count the minimum amount of continuous palindromic subsequences in sequence?
Let $S$ be your input sequence and denote by $S[i:j]$ the subsequence going from the $i$-th to the $j$-th element of $S$ (inclusive).

Let $OPT[i]$ be the minimum amount of continuous palindromic subsequences in which $S[1:i]$ can be partitioned. Let $OPT[0]=0$. For $j=1,\dots,|S|$, you have:

$$
OPT[j] = \min_{\substack{i=1,\dots,j\\S[i:j] \text{ is a palindrome}}} \{1+ OPT[i-1] \}.
$$

The value you are looking for is $OPT[|S|]$. This algorithm requires time $O(|S|^2)$. Notice that all the pairs $(i,j)$ for which $S[i:j]$ is palindrome can be precomputed in $O(|S|^2)$ time.