If you have cleared either the RSTDISBL or the DWEN bit in the configuration fuses, or you have somehow managed to set the SPIEN bit then you need a high voltage programmer to perform any further programming operations. There is no way around this. The AVRISP is not a high voltage programmer, so will not help in this case.
If the configuration fuses still permit low voltage programming, then it shouldn't matter what you do to the SPI programming pins within your ATTiny code, since all programming operations are performed under reset. It's possible that the ISP code you are running on your Arduino is not written very well; the programmers data and clock lines should remain tri-stated until the programmer asserts the reset signal on the chip it's programming.