Does the Maha Parinirvana Sutra acknowledge the existence of eternal spirit/soul? The Buddhist scholar Sallile King has the following interpretation made from the Mahayana Maha PariNirvana Sutra:


  [I]t is obvious that the Mahaparinirvana Sutra does not consider it impossible for a Buddhist to affirm an atman provided it is clear what the correct understanding of this concept is, and indeed the sutra clearly sees certain advantages in doing so.


What I basically want to know is in the context of mindfulness meditation (anapanasati): Is it okay to have the usual hindu belief in eternal aatman for the practice of anapanasati, or is it necessary to have the buddhist belief in non-existence of aatman in order to practice anapanasati?
The second part in the posting below attempts to give an answer to your question stated in the title; the second question I can only comment with subjective view in the first part

1) I think there is no "restriction" or prerequisite (like for instance a system of permits) for mindfulness-meditation. Just try and see how far you come.         

However, the Buddha said, (and it is quite reasonable for me) that for the full liberation from the attachments to emotions and mental figurations, and for the true emancipation from (and eradication of) tanha , the illusion of an inherent, eternal, unchangeable Self atma has necessarily to be dissolved.    



2) I copy&amp;paste some excerpt from the Mahayana-Mahaparinirvana-Sutra (taken from links found at Dr. Tony Page's site). The following are the last couple of paragraphs from the third chapter.
I find it a bit strange to say the least, for instance that a man like we know it as the Buddha from the palicanon comes to be a doctor to "subdue the tirthikas" ...      

If I understand that excerpt correctly, then it is suggested, that the concept of no-self, of "an-atman" is only teached in the framework of concurrence of the "doctors" and "... because of this,(...)" and it is continued, that it had also been said, that there is a self, namely "... it is not that they [the dharma/the phenomena] are completely devoid of a self".       

The text is suggesting, that the Buddha did not teach the non-existence of the self simply because of the non-existence (what would be the natural assumption) but because of didactical reasons (and actually means something else)!   

Below now the excerpt, emphasis added by me: source                         


  Chapter 3
   (... final paragraphs)                                     
  
  "Know, O you Bhiksus!     
  
  The same is the case with the Tathagata, the Alms-deserving, the All-Enlightened-One, the Unsurpassed Best Trainer, the Teacher-of-Heaven-and-Earth, the Buddha-World-Honoured One. He comes as a great Doctor and subdues all tirthikas and bad doctors. In the presence of kings and all people, he says: "I shall become the King of doctors and subdue tirthikas."     
  
  Thus we say: "There is no self, no man, no being, no life, no nurturing, no knowing, none that does, and none that receives."     
  
  O Bhiksus! Know that what the tirthikas say is like the case of a worm that eats upon [a pie-ce of] wood, from which, by chance, there appears what looks like a letter.     
  
  Because of this , the Tathagata teaches and says no-self.     
  
  This is to adjust beings and because he is aware of the occasion. Such non-self is, as occasion arises, spoken of, and it is [also] said that there is the Self.     
  
  This is as in the case of the learned Doctor, who knows well the medicinal and non-medicinal qualities of milk. It is not as with common mortals, who might measure the size of their own self. Common mortals and the ignorant may measure the size of their own self and say, 'It is like the size of a thumb, like a mustard seed, or like the size of a mote.'     
  
  When the Tathagata speaks of Self, in no case are things thus.     
  
  That is why he says:     
  
  'All things have no Self.'      
  
  Even though he has said that all phenomena [dharmas] are devoid of the Self, it is not that they are completely/ truly devoid of the Self.     
  
  What is this Self? Any phenomenon [dharma] that is true [satya], real [tattva], eternal [ni-tya], sovereign/ autonomous/ self-governing [aisvarya], and whose ground/ foundation is un-changing [asraya-aviparinama], is termed 'the Self' [atman].     
  
  This is as in the case of the great Doctor who well understands the milk medicine.     
  
  The same is the case with the Tathagata.     
  
  For the sake of beings, he says "there is the Self in all things"      
  
  O you the four classes! Learn Dharma thus!"