The most important rule for street safety in Tegucigalpa is to never walk anywhere after dark. Are there areas of the city that are safe to walk in after dark? Yes. As an assumedly short-term traveler, do you know what they are? No.
In general, no one in Honduras will intervene during a crime. They do not want to get involved and reap the anger of the perpetrator. They will look the other way and walk right on by. Take special care at night. It is common for a foreigner to be robbed on the streets of Tegucigalpa at night. Thieves will stake out areas near tourist hotels, especially the Hotel Maya.
Cars are commonly broken into in broad daylight and the thieves don't even bother wearing masks. If you are driving, it is always worth it to pay to park in a guarded lot.
Follow these general guidelines:
Keep to the main parts of the city and don't be tempted to go to places that you are not sure of.
Keep to the main roads and avoid short cuts down back alleys etc.
Never walk at night in the center of the city even for a short distance - always take a taxi.
Be particularly wary of people hanging around outside hotels; it is a favorite place to catch tourists and mug them.
Ignore the street children and people coming up to you in the streets with hard luck tales. Street children can become violent and the latter may be part of an elaborate scam or they might just simply be pick pockets. The best thing to do is just to walk on and ignore them.
Do not carry large sums of money when shopping and do not wear expensive jewelry.
Do not accept food and drink from strangers; visitors have known to be drugged and then robbed.
If you must carry large sums of money or valuable possessions, carry two wallets: Keep one hidden with most of your money in it. The other should be in the most common place, your back pocket. Keep 5-10 US$ in the wallet, and a few stray lempiras. The lempiras can go to beggars they tend to be persistent, and the dollars to appease any possible robbers. Typically 5-10 US$ is viewed as a days salary in Honduras, and just may be enough to appease a robber without sacrificing your larger stash. Use caution, as there is no such thing as a predictable thief in Honduras.
If you shop at the Mercado San Isidro in Comayagüela, don't go after dark and don't carry a lot of valuables with you. Even in the daytime there are pickpockets and "grab-and-run" thieves in the market. And definitely don't walk around in Comayagüela itself apart from the market area at ANY time, day or night.
It IS possible to go to Parque de la Paz and take panoramic pictures from the top as well as a picture of the monument itself. However, you must be very careful. Do not ever walk up the hill by yourself or even in a small group. Have a reliable taxi or tourist guide drive you up and make sure not to wander around too much when you're at the summit. It ought to go without saying that this is only possible in the daytime.