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JavaScript (ES6), 63 bytes

(x,y,a)=>a.map(P=(v,i)=>v-y?v-x?0:a=i:1/(p=a)?P=+P||i:0)&&[P,p]

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Commented

(x, y, a) =>          // given the two integers x, y and the array a[]
  a.map(P =           // initialize P to a non-numeric value
            (v, i) => // for each value v at position i in a[]:
    v - y ?           //   if v is not equal to y:
      v - x ?         //     if v is not equal to x:
        0             //       do nothing
      :               //     else (v = x):
        a = i         //       save the current position in a
    :                 //   else (v = y):
      1 / (p = a) ?   //     update p to a (last position of x); if p is numeric (>= 0):
        P = +P || i   //       unless P is also already numeric, update it to i
                      //       (if P is numeric, it's necessarily greater than 0 because
                      //       we've also seen x before; that's why +P works)
      :               //     else:
        0             //       do nothing
  )                   // end of map()
  && [P, p]           // return [P, p]

Alternate versions

Using JS built-ins, a more straightforward answer is 79 bytes:

(x,y,a)=>[a.indexOf(y,a.indexOf(x)),a.slice(0,a.lastIndexOf(y)).lastIndexOf(x)]

which can be slightly compressed to 75 bytes:

(x,y,a)=>[a.indexOf(y,a.indexOf(x)),a.slice(0,a[L='lastIndexOf'](y))[L](x)]

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Edit: @Neil managed to reduce it to a very nice 67-byte:

(x,y,a,f=s=>a[z=y,y=x,x=z,s+=`ndexOf`](x,a[s](y)))=>[f`i`,f`lastI`]

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APL (Dyalog Classic), 29 27 bytes

⊃{(⊃⍵~⍳⊃⍺),⊃⌽⍺∩⍳⊃⌽⍵}/⍸¨⎕=⊂⎕

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prompts for the array and then for var1,var2

Python 2, 72 bytes

def f(x,y,a):i=a.index;j=a[::-1].index;print i(y,i(x)),len(a)+~j(x,j(y))

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