I solved your system in Mathematica. But I can suggest some errors in your system. First of all, you didn't specify all variables. Second, you wrote (a2x1) that may be interpreted as a variable, not a product, i.e. you must write it as (a2 x1).

Did you mean these ones?

In[10]:= Solve[{a1 x0^2 + b1 x0 + c1 == y0, 
  a1  x1 ^2 + b1 x1 + c1 == y1, a2  x1^2 + b2 x1 + c2 == y1, 
  a2 x2^2 + b2 x2 + c2 == y2, 2 a1 x0 + b1 == 0, 2 a2 x2 + b2 == 0, 
  2 a2  x1 + b2 == 2 a2  x1 + b2}, {a1, a2, b1, b2, c1, c2, x0, y0, 
  x1, y1, x2, y2}]

During evaluation of In[10]:= Solve::svars: Equations may not give solutions for all "solve" variables. >>

Out[10]= {{b1 -> -2 a1 x0, b2 -> -2 a2 x2, 
  c2 -> c1 - 2 a1 x0 x1 + a1 x1^2 - a2 x1^2 + 2 a2 x1 x2, 
  y0 -> c1 - a1 x0^2, y1 -> c1 - 2 a1 x0 x1 + a1 x1^2, 
  y2 -> c1 - 2 a1 x0 x1 + a1 x1^2 - a2 x1^2 + 2 a2 x1 x2 - a2 x2^2}}

In[11]:= Reduce[{a1 x0^2 + b1 x0 + c1 == y0, 
  a1  x1 ^2 + b1 x1 + c1 == y1, a2  x1^2 + b2 x1 + c2 == y1, 
  a2 x2^2 + b2 x2 + c2 == y2, 2 a1 x0 + b1 == 0, 2 a2 x2 + b2 == 0, 
  2 a2  x1 + b2 == 2 a2  x1 + b2}, {a1, a2, b1, b2, c1, c2, x0, y0, 
  x1, y1, x2, y2}]

Out[11]= (a1 == 0 && a2 == 0 && b1 == 0 && b2 == 0 && c2 == c1 && 
   y0 == c1 && y1 == c1 && y2 == c1) || (a1 == 0 && b1 == 0 && 
   y0 == c1 && 
   a2 != 0 && (x1 == (-b2 + Sqrt[b2^2 + 4 a2 c1 - 4 a2 c2])/(2 a2) || 
     x1 == -((b2 + Sqrt[b2^2 + 4 a2 c1 - 4 a2 c2])/(2 a2))) && 
   y1 == c1 && x2 == -(b2/(2 a2)) && 
   y2 == 1/2 (2 c2 + b2 x2)) || (a2 == 0 && b2 == 0 && a1 != 0 && 
   x0 == -(b1/(2 a1)) && 
   y0 == 1/2 (2 c1 + b1 x0) && (x1 == (-b1 - Sqrt[
       b1^2 - 4 a1 c1 + 4 a1 c2])/(2 a1) || 
     x1 == (-b1 + Sqrt[b1^2 - 4 a1 c1 + 4 a1 c2])/(2 a1)) && 
   y1 == c2 && y2 == c2) || (a2 == a1 && b2 == b1 && c2 == c1 && 
   a1 != 0 && x0 == -(b1/(2 a1)) && y0 == 1/2 (2 c1 + b1 x0) && 
   y1 == c1 + b1 x1 + a1 x1^2 && x2 == x0 && 
   y2 == 1/2 (2 c1 + b1 x0)) || (a2 == a1 && a1 != 0 && 
   x0 == -(b1/(2 a1)) && y0 == 1/2 (2 c1 + b1 x0) && b1 - b2 != 0 && 
   x1 == (-c1 + c2)/(b1 - b2) && y1 == c1 + b1 x1 + a1 x1^2 && 
   x2 == -(b2/(2 a1)) && y2 == 1/2 (2 c2 + b2 x2)) || (a1 != 0 && 
   x0 == -(b1/(2 a1)) && y0 == 1/2 (2 c1 + b1 x0) && 
   a1 - a2 != 
    0 && (x1 == (-b1 + b2 - 
       Sqrt[(b1 - b2)^2 - 4 (a1 - a2) (c1 - c2)])/(2 (a1 - a2)) || 
     x1 == (-b1 + b2 + Sqrt[(b1 - b2)^2 - 4 (a1 - a2) (c1 - c2)])/(
      2 (a1 - a2))) && y1 == c1 + b1 x1 + a1 x1^2 && a2 != 0 && 
   x2 == -(b2/(2 a2)) && y2 == 1/2 (2 c2 + b2 x2))