Is there a way to run differential equations to a specific point and then change input values, continue running, and graph it together?

The issue is that you used the argument callbacks instead of callback. That was my bad (and we really need to error for that). A corrected code is:

using DifferentialEquations

f = @ode_def_bare ManagedLinear begin
  dBB = r*(1- BB/K)*BB -xI*yIB*BI*((BB^h/(B0^h + BB^h))/eIB)
  dBI = -xI*BI + xI*yIB*BI*(BB^h/(B0^h + BB^h))-xT *yTI*BT*((BI^h/(B0^h + BI^h))/eTI)
  dBT = -xT*BT + xT*yTI*BT*(BI^h/(B0^h + BI^h))-q*BT*E + f1 + f2
end K r yIB yTI eIB eTI B0 h xI xT q E f1 f2

function condition(u,t,integrator)
  t == 200.
end
function affect!(integrator)
  integrator.p[13] = 0.01
end
cb = DiscreteCallback(condition, affect!)

function condition2(u,t,integrator)
  t == 200.
end
function affect2!(integrator)
  integrator.p[14] = 50
end
cb2 = DiscreteCallback(condition2, affect2!)

cbs = CallbackSet(cb,cb2)

u0 = [500.0,200.0,100.0]
tspan = (0.0,1000.0)
p = [450, 1.1, 10, 10, 0.66, 0.85, 80, 1.2, 0.15, 0.06, 0.01, 50, 0, 0] # initial f1 = 0.0 f2 = 0.0
# K r yIB yTI eIB eTI B0 h xI xT q E
prob = ODEProblem(f,u0,tspan,p)
sol=solve(prob,Rosenbrock23(),callback=cbs,tstops=[200.0],reltol=1e-6)

using Plots

plot(sol, title = "Food Chain Managed Linear ", xlabel = "Time" ,ylabel = "Density", lw=0.5)

plot