There are some very low-odour paraffins but they are expensive. Some lamp oils, and some heater oils are odourless. There is one posh electrically-controlled paraffin heater available that uses its own fuel which you can buy in 5l+ quantities and is a bit cheaper
Found a source of lamp oil in the garden ornament section of BinQ. Not on shelves at moment, no doubt hidden away to further the onslaught of the tupperware xmas tree. Appears to be stocked for summer oil lamps in garden, virtually odourless.
I bought a Tilley new and had endless trouble with it, so sent it back and got a Vapalux M320 off Ebay instead. Seems like a much better product all round.
I've found that the only time I get the paraffin smell is when I turn it off at the end of using it... if I take it outside to turn it off before bringing it inside again, zero odour.
NB. You can't run a Tilley or any other paraffin pressure lamp on lamp oil I don't think.
Lamp oil is straight paraffin without additives and low in sulphur. It is perfectly OK with a pressure stove. There are toxicity problems with coloured lamp oils, they are banned in Germany for example. I guess it is possible that the dyes might lead to deposition in the jet but otherwise no problems should occur.
White spirit and paraffin are interchangeable for most purposes. White spirit is, I'm told a narrower cut fraction compared with paraffin. I've used paraffin to dilute oil paints and lamp oil to wipe away unset Sikaflex.
Concerning the quality of Tilley lamps: it's been said that in 1957 they sacked their last engineer and promoted an accountant instead. The one to get is the pre '54 Tilley Guardsman.