Does PETSc really give speedup?
You misread the text.  The authors of PETSc are just telling you that you can't avoid Amdahl's law.

They have done their best to parallelize every aspect of the linear solver.  But a real program is not just a call to a linear solver.  First you generate a matrix and then you pass the matrix to the linear solver.  If your matrix generator is slow, your whole program will be slow.

For example, suppose your original program spends 1000 seconds generating the matrix $A$ and vector $b$ and then you call a linear solver.  Your old (sequential) linear solver took 1000 seconds to find $x$ such that $Ax = b$.  Now you replace your old sequential linear solver with PETSc.  Suppose the PETSc authors did such a good job that the PETSc parallel linear solver finds $x$ in just 1 second!  Now how long does it take your program to run?  1001 seconds.  You got less than 2x speedup!  You need to do some work on your matrix generation code if you want to get a better speedup.

Pretty much the authors of PETSc are just telling you that a parallelized linear solver library is not a magic bean.