We present a novel method based on optical fibre tapering for fabrication of Surface Nanoscale Axial Photonics (SNAP) devices with parabolic profiles with an unprecedentedly large number of axial eigenmodes. Tapering of a commercial 125 μm single-mode optical fibre to a 30 μm diameter waist by laser brushing creates a SNAP bottle microresonator with parabolic radius variation in the centre of the tapered region. Ideal parabolic resonators should demonstrate equal spacing between resonances. Our spectral measurement of the parabolic profile shows spacing of ∼6 GHz with 10% deviation over a bandwidth of 2.5 THz containing up to 400 axial eigenfrequencies. This new discovery for the creation of SNAP parabolic microresonator devices is important for fabrication of miniature delay lines, buffers and frequency comb generators. Characterisation of our exemplar microresonators is briefly explored, particularly for broadband frequency comb generators which require equidistant frequency spacing. Further investigations include scaling of the parabolic feature with tapering process parameters, repeatability testing, and the fabrication of more complex shapes.