Parallelize[eqn = Piecewise[{{-\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(g = p + x\),
SubscriptBox[\(p\), \(0\)]]\(2
\*SubscriptBox[\(K\), \(f\)]
\(\*SubscriptBox[\(N\), \(g\)]\)[t]\)\),
Subscript[p, 0] - x <= p <= Subscript[p, 0]}, {0,
x <= p <=
Subscript[p, 0] - x}, {+Subscript[K,
f]*(p - x + 1) Subscript[N, p][t], 0 <= p <= x - 1}}]
sol =
DSolve[Derivative[1][Subscript[N, p]][t] +
Subscript[K, f]*(p - x + 1) Subscript[N, p][t] - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(g = p + x\),
SubscriptBox[\(p\), \(0\)]]\(2
\*SubscriptBox[\(K\), \(f\)]
\(\*SubscriptBox[\(N\), \(g\)]\)[t]\)\) == eqn, Subscript[N, p][t],
t]]

1. 为什么我的代码加了下标（Subscript）就出问题了？ https://tieba.baidu.com/p/6149871454
2. 【讨论】并行到底有多难 https://tieba.baidu.com/p/6149871454
3. sol 前方缺少分号，换行被识别为乘法
4. N和K在System`上下文中已被声明，你应该换个名字